Research ArticlePerformance and Complexity Evaluation of Iterative Receiverfor Coded MIMO-OFDM Systems
Rida El Chall1 Fabienne Nouvel1 Maryline Heacutelard1 and Ming Liu2
1 INSA IETR CNRS UMR 6164 35708 Rennes France2Beijing Key Lab of Transportation Data Analysis and Mining Beijing Jiaotong University Beijing 100044 China
Correspondence should be addressed to Rida El Chall ridael-challinsa-rennesfr and Ming Liu mingliubjtueducn
Received 17 July 2015 Revised 25 September 2015 Accepted 17 December 2015
Academic Editor Yuh-Shyan Chen
Copyright copy 2016 Rida El Chall et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Multiple-input multiple-output (MIMO) technology in combination with channel coding technique is a promising solution forreliable high data rate transmission in future wireless communication systems However these technologies pose significantchallenges for the design of an iterative receiver In this paper an efficient receiver combining soft-input soft-output (SISO) detectionbased on low-complexity K-Best (LC-K-Best) decoder with various forward error correction codes namely LTE turbo decoderand LDPC decoder is investigated We first investigate the convergence behaviors of the iterative MIMO receivers to determinethe required inner and outer iterations Consequently the performance of LC-K-Best based receiver is evaluated in various LTEchannel environments and comparedwith otherMIMOdetection schemesMoreover the computational complexity of the iterativereceiver with different channel coding techniques is evaluated and compared with different modulation orders and coding ratesSimulation results show that LC-K-Best based receiver achieves satisfactory performance-complexity trade-offs
1 Introduction
The ever increasing demand for higher data rate and betterlink reliability poses challenges for the modern wireless com-munication systems such as IEEE 80211 80216 DVB-NGH3GPP long term evolution (LTE) and LTE-Advanced (LTE-A) The combination of multiple antennas at transmitterandor receiver orthogonal frequency-division multiplexing(OFDM) technique state-of-the-art channel coding schemesand iterative reception techniques has been seen as thepromising solution for the future wireless systems
MIMO technology which utilizes multiple antennas attransmitter andor receiver is able to achieve high diversitythrough space-time coding and high data rate through spatialmultiplexing [1] It is commonly used in combination withOFDM technique to combat intersymbol interference (ISI)and therefore achieve better spectral efficiency Modernchannel coding schemes such as turbo codes or LDPCcodes are powerful forward error correction (FEC) codesthat are able to protect the integrity of the transmitted dataand to approach the channel capacity Therefore the codedMIMO-OFDM systems are recognized as attractive solutions
for the future high speed wireless communication systemsHowever the practical design of such coded MIMO-OFDMsystems involves numerous challenges at the receiver
The reception strategy that offers best performance is tojointly detect and decode the received symbols However thisjoint detection scheme has been shown to be very complexand infeasible for practical implementation [2] Alternativelythe optimal performance can be approached by the iterativeprocessing or commonly referred to as turbo processing [3ndash6]which replaces the joint detection by iteratively performingindependent detection and decoding processing It consists ofsoft-input soft-output (SISO) detector and channel decoderthat exchange ldquosoftrdquo information [7]
Regarding theMIMO detectionmethod the optimal wayrelies on maximum a posteriori probability (MAP) algo-rithm However it presents a complexity that exponentiallyincreases with respect to the number of transmit antennasand modulation orders Hence several suboptimal but low-complexity detectors have been proposed in the literatureThese solutions include the family of linear equalizer inter-ference canceller and tree-search detector To achieve betterperformance the design and the implementation of SISO
Hindawi Publishing CorporationMobile Information SystemsVolume 2016 Article ID 7642590 22 pageshttpdxdoiorg10115520167642590
2 Mobile Information Systems
MIMO detectors have been also widely investigated such asthe minimum mean square error-interference cancellation(MMSE-IC) [8 9] improved VBLAST (I-VBLAST) [10 11]list sphere decoder (LSD) [12] single tree-search spheredecoder (STS-SD) [13ndash15] K-Best decoder [16ndash20] and fixedsphere decoder (FSD) [21ndash23] Among them MMSE-IC andI-VBLAST present low computational complexity but theyare not able to fully exploit the spatial diversity of MIMOsystem Meanwhile the sphere decoder is able to achievesuperior performance However the sphere decoder uses adepth-first searchmethodTherefore its computational com-plexity varies significantly with respect to the channel con-dition yielding prohibitive worst-case complexity Moreoverthe sphere decoder suffers from variable throughput due toits sequential tree-search strategy which makes it unsuitablefor parallel implementation In contrast the breadth-firstsearch based K-Best and FSD algorithms are hence moreattractive for practical implementation than sphere decodingas they can offer stable throughput at a cost of acceptableperformance loss
Despite these efforts it is still very challenging to developa high speed iterative MIMO receiver to meet the highthroughput requirements of future wireless communicationsystems at affordable complexity and implementation costIn [24] the performance-complexity trade-offs of iterativeMIMO receiver have been investigated However the investi-gation is limited to the turbo channel coding and theoreticalchannel cases In this contribution the performance andthe complexity of iterative MIMO receiver are evaluated ina much broader and more practical scope We investigatein depth the soft joint iterative detection schemes withvarious symbol detection schemes various soft-input soft-output channel decoders and various ways of constructingjoint loops under different channel conditions In particularthe most representative modern channel coding schemesincluding LTE turbo code and LDPC code are consideredSeveral LTE multipath channel models are employed in thesimulation to evaluate the performance in real propagationscenarios Consequently a detailed comparative study is con-ducted among iterative receivers with different modulationsand channel coding schemes (turbo LDPC) It has beendemonstrated through the comparison that LC-K-Best basedreceiver achieves a best trade-off between performance andcomplexity among the iterative MIMO receivers consideredin this work
The remainder of this paper is organized as followsSection 2 presents the MIMO-OFDM system model andthe concept of iterative detection-decoding process Channeldecoding based on turbo decoder and LDPC decoder isdescribed in Section 3 Section 4 briefly reviews the mostrelevant SISO MIMO detection algorithms based on spheredecoder LC-K-Best decoder and interference canceller InSection 5 the convergence behavior of the iterative receiversis discussed using extrinsic information transfer (EXIT) chartto retrieve to required number of inner and outer itera-tions Section 6 illustrates the performance of our proposedapproaches in LTE-based channel environments Then thecomputational complexity of the receivers with both turboand LDPC coding techniques is evaluated and compared
with different modulation orders and coding rates Section 7concludes the paper
2 System Model
21 MIMO-OFDM System Model We consider a MIMO-OFDM system based on bit-interleaved coded modulation(BICM) scheme [25] with 119873
119905transmit antennas and 119873
119903
receive antennas (119873119903
ge 119873119905) as depicted in Figure 1
At the transmitter the information bits of length 119870119887are
first encoded by a channel encoder which outputs a codewordc of length 119873
119887with a coding rate 119877
119888= 119870
119887119873
119887 The channel
encoder can be a turbo encoder or an LDPC encoder Theencoded bits are then randomly interleaved andmapped intocomplex symbols of 2
119876 quadrature amplitude modulation(QAM) constellation where 119876 is the number of bits persymbol The symbols are mapped into 119873
119905transmit antennas
using either space-time block coding (STBC) schemes orspatial multiplexing (SM) schemes offering different diversitygain and multiplexing gain trade-offs Herein the SM-basedMIMO system is considered without loss of generality IFFTis applied to 119873
119888parallel symbols to obtain the time domain
OFDM symbols where 119873119888is the number of useful subcarri-
ers The symbols are then sent though the radio channel afterthe addition of the cyclic prefix (CP) which is assumed largerthan the maximum delay spread of the channel The timedomain symbol transmitted by the 119894th antenna is expressedas
119904119894(119899) =
1
radic119873FFT
119873FFTminus1
sum
119896=0
119878119894(119896) 119890
1198952120587119896119899119873FFT
minus119873119892
le 119899 le 119873FFT minus 1
(1)
where 119878119894(119896) is the symbol in the frequency domain before
IFFT 119873FFT is the size of the FFT and 119873119892is the length of
the CP The transmit power is normalized so that Ess119867 =
119864119904119873
119905I119873119905
where I119873119905
is the 119873119905
times 119873119905identity matrix The
transmission information rate is 119877119888
sdot 119873119905
sdot 119876 bits per channeluse
Using the OFDM technique the frequency-selective fad-ing channel is divided into a series of orthogonal and flat-fading subchannels The signal equalization is performed bya simple one-tap equalizer at the receiver Therefore after theremoval of CP FFT is performed to get the frequency domainsignal vector y
119896= [119910
1 1199102 119910
119873119903
]119879 that can be expressed as
y119896
= H119896s119896
+ n119896 (2)
where 119896 = 1 119873119888is the index of subcarriers For simplicity
the subcarrier index 119896 is omitted in the sequel H is the119873119903
times 119873119905channel matrix with its (119894 119895)th element ℎ
119894119895 the
channel frequency response of the channel link from 119895thtransmit antenna to 119894th receive antenna The coefficients ofthe channel matrix H are assumed to be perfectly known atthe receiver n = [119899
1 1198992 119899
119873119903
]119879 is the independent and
identically distributed (iid) additive white Gaussian noise(AWGN) vector with zero mean and variance of 119873
0= 120590
2
119899
Mobile Information Systems 3
Receiver
Transmitter Mapping
QAM MIMOmapper
Channel encoder
Channel decoder
DemapperQAM
MIMOdetector
Softmapper
IFFT
IFFT
FFT
FFT
u c e x s
H
y
LA1
LE1
Iout
LA2
LE2
Iin
Π
Π
Πminus1u xc
Figure 1 Block diagram of MIMO-OFDM system using bit-interleaved coded modulation with iterative detection and decoding
22 Iterative Detection-Decoding Principle At the receiver torecover the transmitted signal from interferences an iterativedetection-decoding process based on the turbo principle isapplied as depicted in Figure 1 The MIMO detector andthe channel decoder exchange soft information that is loglikelihood ratio (LLR) in each iteration
The MIMO detector takes the received symbol vector yand the a priori information 119871
1198601of the coded bits from the
channel decoder and computes the extrinsic information1198711198641
The MIMO detection algorithm can be the MAP algorithmor other suboptimal algorithms like STS-SDK-Best decoderI-VBLAST or MMSE-IC The extrinsic information is dein-terleaved and becomes the a priori information 119871
1198602for the
channel decoderThe channel decoder computes the extrinsicinformation 119871
1198642that is reinterleaved and fed back to the
detector as the a priori information 1198711198601
The channel decoding is performed either by an LTEturbo decoder or by an LDPC decoder which exchanges softinformation between their component decoders as describedin the next section In our iterative process we denote thenumber of outer iterations between the MIMO detector andthe channel decoder by 119868out and the number of iterationswithin the turbo decoder or LDPC decoder by 119868in
For QAM the mapping process can be done inde-pendently for real and imaginary part The system modelexpressed in (2) can be converted into an equivalent real-valued model
[
Re (y)
Im (y)] = [
Re (H) minus Im (H)
Im (H) Re (H)] [
Re (s)Im (s)
]
+ [
Re (n)
Im (n)]
(3)
where Re(sdot) and Im(sdot) represent the real and imaginary partsof a complex number respectively Each QAM constellationpoint is treated as two PAM symbols and the matrix dimen-sion is doubled However as shown in [17] the real-valuedmodel is more efficient for the implementation of the sphere
decoder Hence it will be used as the systemmodel in case ofsphere decoding in the following sections
3 Soft-Input Soft-Output Channel Decoder
Channel coding is used to protect the useful informationfrom channel distortion and noise by introducing someredundancy The state-of-the-art channel coding schemessuch as the LDPC [26] and turbo codes [3] can effectivelyapproach the Shannon bound LDPC codes are nowadaysadopted in many standards including IEEE 80211 and DVB-T2 as they achieve very high throughput due to inherentparallelism of the decoding algorithm In the meantime theturbo codes are also adopted in LTE LTE-A (binary turbocodes) and WiMAX (double binary turbo codes) In thispaper LDPC codes and LTE turbo codes are considered
31 Turbo Decoder Initially proposed in 1993 [3] turbocodes have attracted great attention due to the capacity-approaching performance The turbo encoder is constitutedby a parallel concatenation of two recursive systematicconvolutional encoders separated by an interleaver Thefirst encoder processes the original data while the secondprocesses the interleaved version of dataThemain role of theinterleaver is to reduce the degree of correlation between theoutputs of the component encoders
In LTE system the recursive systematic encoders with8 states and [13 15]
119900polynomial generators are adopted A
quadratic polynomial permutation (QPP) interleaver is usedas a contention free interleaver and it is suitable for paralleldecoding of turbo codes as illustrated in Figure 2(a) Themother coding rate is 13 Coding rates other than themotherrate can be achieved by puncturing or repetition using the ratematching technique
The turbo decoding is performed by two SISO compo-nent decoders that exchange soft information of their datasubstreams Each component decoder takes systematic orinterleaved information the corresponding parity informa-tion and the a priori information from the other decoder to
4 Mobile Information Systems
LTE QPPinterleaver
D D D
DDD
uk (Info)
1st encoder
2nd encoder
sk
p1k
p2k
s998400k
⨁
⨁ ⨁
⨁
⨁
⨁
⨁
⨁
(a) Turbo encoder
DEC2
DEC1
Lc(in) SP
Lc(p1)
Lc(s)
Lc(p2)
Πi
La(u)
Πminus1i ΠiIin
Le(u)
Le(u)
Le(u)
La(u)
Le(p2)
Le(p1)
PSLc(out)
(b) Turbo decoder
Figure 2 Structure of LTE turbo code with 119877119888
= 13 (a) turboencoder and (b) turbo decoder
compute the extrinsic information as shown in Figure 2(b)Two families of decoding algorithms can be used soft-output Viterbi algorithms (SOVA) [27 28] and maximum aposteriori (MAP) algorithm [29] The MAP algorithm offerssuperior performance but suffers from high computationalcomplexity Two suboptimal algorithms namely log-MAPand max-log-MAP are practically used [30] Herein log-MAPalgorithm is considered by using the Jacobian logarithm[30]
where 119891119888(|119909 minus 119910|) is a correction function that can be
computed using a small look-up table (LUT)The decoder computes the branch metrics (120574) and the
forward (120572) and the backward (120573) metrics between two statesin the trellis as follows
120574119896
(119904119896minus1
119904119896) = 119901 (119904
119896 119910119896
| 119904119896minus1
)
120572119896
(119904119896) =
lowastmax119904119896minus1
(120572119896minus1
(119904119896minus1
) + 120574119896
(119904119896minus1
119904119896))
120573119896
(119904119896) =
lowastmax119904119896+1
(120573119896+1
(119904119896+1
) + 120574119896
(119904119896 119904119896+1
))
(5)
The a posteriori LLRs of the information bits are computed as
119871 (119906119896) =
lowastmax119906119896=0
(120572119896minus1
(119904119896minus1
) + 120574119896
(119904119896minus1
119904119896) + 120573
119896(119904119896))
minuslowastmax119906119896=1
(120572119896minus1
(119904119896minus1
) + 120574119896
(119904119896minus1
119904119896) + 120573
119896(119904119896))
(6)
The component decoders exchange only the extrinsic LLRwhich is defined by
119871119890(119906119896) = 119871 (119906
119896) minus 119871
119886(119906119896) minus 119871
119904(119906119896) (7)
where 119871119886(119906119896) and 119871
119904(119906119896) correspond to the a priori informa-
tion from the other decoder and the systematic informationbits respectively
32 LDPC Decoder LDPC codes belong to a class of linearerror correcting block codes first proposed by Gallager [26]Their main advantages lie in their capacity-approaching per-formance and their low-complexity parallel implementations[31] LDPC codes can be represented by a parity checkmatrix HLDPC or intuitively through Tanner graph [32]Tanner graphs are bipartite graphs containing two types ofnodes the check nodes and the variable nodes as illustratedin Figure 3(a) It consists of 119872 check nodes (CN) whichcorrespond to the number of parity bits (ie number of rowsof HLDPC) and 119873 variable nodes (VN) corresponding to thenumber of bits in a codeword (ie number of columns ofHLDPC)
The optimal maximum a posteriori decoding of LDPCcodes is infeasible from the practical implementation point ofviewAlternatively LDPCdecoding is done using themessagepassing or belief propagation algorithms which iterativelypass messages between check nodes and variable nodes asshown in Figure 3(b) The belief propagation is denotedas the sum-product decoding because probabilities can berepresented as LLRs which allow the calculation of messagesusing sum and product operations
Let 119871V119894119895
be themessage from variable node 119894 to check node119895 and 119871
119888119895119894
the message from check node 119895 to variable node119894 Let 119881
119895and 119862
119894denote the set of adjacent variable nodes
connected to the check node 119895 and the set of adjacent checknodes connected to the variable node 119894 respectively where0 le 119895 le 119872 and 0 le 119894 le 119873 For the first iteration theinput to the LDPC decoder is the LLRs 119871(119888
119894) of the codeword
c which are used as an initial value of the extrinsic variablenode messages that is 119871V
119894119895
= 119871(119888119894) For the 119896th iteration the
algorithm can be summarized as follows(1) Each check node 119895 computes the extrinsic message to
its neighboring variable node 119894
119871119888119895119894
= 2tanhminus1 ( prod
1198941015840isin119881119895119894
tanh(
119871V1198941015840119895
2)) (8)
(2) Each variable node 119894 updates its extrinsic informationto the check node 119895 in the next iteration
119871V119894119895
= 119871 (119888119894) + sum
1198951015840isin119862119894119895
1198711198881198951015840119894
(9)
Mobile Information Systems 5
n = 6m = 4
VN (n)
CN (m)
0 1 1 0 0 1
1 1 1 0 1 0
1 0 0 1 1 1
0 0 1 1 0 1
=HLDPC
(a)
1 2 3 5 64
1 2 3 4
VNi
CNj
L13
L12
L31
L32
L34
Lc12Lc13
Lc16Lc31
Lc34
Lc35
Lc36
(b)
Figure 3 LDPC decoder (a) matrix representation and Tanner graph and (b) iterative decoding between VNs and CNs
(3) The a posterioriLLRof each codeword bit is computedas
119871119901
(119888119894) = 119871 (119888
119894) + sum
1198951015840isin119862119894
1198711198881198951015840119894
(10)
The decoding algorithm alternates between check node proc-essing and variable node processing until a maximum num-ber of iterations are achieved or until the parity check con-dition is satisfied When the decoding process is terminatedthe decoder outputs the a posteriori LLR
4 Soft-Input Soft-Output MIMO Detection
The aim of MIMO detection is to recover the transmittedvector s from the received vector y The state-of-the-artMIMO detection algorithms have been presented in [24]These algorithms can be divided into two main familiesnamely the tree-search-based detection and the interference-cancellation-based detection In this section we brieflyreview the main existing SISO MIMO detection algorithmsuseful for the following sections
41 MaximumA Posteriori Probability (MAP) Detection TheMAP algorithm achieves the optimum performance throughthe use of an exhaustive search over all 2
119876sdot119873119905 possible symbol
combinations to compute the LLR of each bit The LLR of the119887th bit in the 119894th transmit symbol 119909
119894119887 is given by
119871 (119909119894119887
) = log119875 (119909
119894119887= +1 | y)
119875 (119909119894119887
= minus1 | y)
= logsumsisin120594+1
119894119887
119901 (y | s) 119875 (s)
sumsisin120594minus1119894119887
119901 (y | s) 119875 (s)
(11)
where 120594+1
119894119887and 120594
minus1
119894119887denote the sets of symbol vectors in
which the 119887th bit in the 119894th antenna is equal to +1 and minus1respectively 119901(y | s) is the conditioned probability densityfunction given by
119901 (y | s) =1
(1205871198730)119873119903
exp(minus1
1198730
1003817100381710038171003817y minus Hs10038171003817100381710038172
) (12)
119875(s) represents the a priori information provided by thechannel decoder in the form of a priori LLRs
119871119860
(119909119894119887
) = log119875 (119909
119894119887= +1)
119875 (119909119894119887
= minus1) forall119894 119887
119875 (s) =
119873119905
prod
119894=1
119875 (119904119894) =
119873119905
prod
119894=1
119876
prod
119887=1
119875 (119909119894119887
)
(13)
The max-log-MAP approximation is commonly used in theLLR calculation with lower complexity [12]
119871 (119909119894119887
) asymp1
1198730
min120594minus1
119894119887
1198891 minus
1
1198730
min120594+1
119894119887
1198891 (14)
1198891
=1003817100381710038171003817y minus Hs1003817100381710038171003817
2
minus 1198730
119873119905
sum
119894=1
119876
sum
119887=1
log119875 (119909119894119887
) (15)
where 1198891represents the Euclidean distance between the
received vector y and lattice pointsHsBased on the a posteriori LLRs 119871(119909
119894119887) and the a priori
LLRs 119871119860
(119909119894119887
) the detector computes the extrinsic LLRs119871119864(119909119894119887
) as
119871119864
(119909119894119887
) = 119871 (119909119894119887
) minus 119871119860
(119909119894119887
) (16)
The MAP algorithm is not feasible due to its exponentialcomplexity since 2
119876sdot119873119905 hypotheses have to be considered
within each minimum term and for each bit Thereforeseveral suboptimal MIMO detectors have been proposedwith reduced complexity as will be briefly discussed in thefollowing sections
42 Tree-Search-Based Detection The tree-search-baseddetection methods generally fall into two main categoriesnamely depth-first search like the sphere decoder andbreadth-first search like the K-Best decoder
421 List Sphere Decoder (LSD) The basic idea of the spheredecoder is to limit the search space of the MAP solution to ahypersphere of radius 119903
119904around the received vector Instead
of testing all the hypotheses of the transmitted signal onlythe lattice points that lie inside the hypersphere are testedreducing the computational complexity [33]
sSD = arg minsisin2119876119873119905
1003817100381710038171003817y minus Hs1003817100381710038171003817
2
le 1199032
119904 (17)
6 Mobile Information Systems
Using the QR decomposition in real-valued model thechannel matrix H can be decomposed into two matrixes Qand R (H = QR) where Q is 2119873
119903times 2119873
119905orthogonal matrix
(Q119867Q = I2119873119905
) and R is 2119873119905
times 2119873119905upper triangular matrix
with real-positive diagonal elements [34] Therefore the dis-tance in (17) can be computed as yminusHs2 = yminusRs2 wherey = Q119867y is the modified received symbol vector Exploitingthe triangular nature of R the Euclidean distance metric 119889
1
in (15) can be recursively evaluated through the accumulatedpartial Euclidean distance (PED) 119889
and the a priori-based partial metric at the 119894th level respec-tively
This process can be illustrated by a tree with 2119873119905
+ 1
levels as depicted in Figure 4(a) The tree search starts at theroot level with the first child node at level 2119873
119905 The partial
Euclidean distance 1198892119873119905
in (18) is then computed If 1198892119873119905
issmaller than the sphere radius 119903
119904 the search continues at level
2119873119905minus1 and steps down the tree until finding a valid leaf node
at level 1List sphere decoder is proposed to approximate the MAP
detector [12] It generates a list L sub 2119876119873119905 that includes the
best possible hypotheses The LLR values are then computedfrom this list as
119871 (119909119894119887
) =1
1198730
minLcap120594minus1119894119887
1198891 minus
1
1198730
minLcap120594+1119894119887
1198891 (19)
The main issue of LSD is the missing counter-hypothesisproblem depending on the list sizeThe use of limited list sizecauses inaccurate approximation of the LLR due to missingsome counter hypotheses where no entry can be found inthe list for a particular bit 119909
119894119887= +1 minus1 Several solutions
have been proposed to handle this issue LLR clipping is afrequently used solution which consists simply to set the LLRto a predefined maximum value [12 35]
Several methods can be considered to reduce the com-plexity of the sphere decoder such as Schnorr-Euchner(SE) enumeration [36] layer ordering technique [34] andchannel regularization [37] Layer ordering technique allowsthe selection of the most reliable symbols at a high layerusing the sorted QR (SQR) decomposition However channelregularization introduces a biasing factor in the metricswhich should be removed in LLR computation to avoidperformance degradation as discussed in [38] In the sequelthe SQR decomposition is considered in the preprocessingstep
422 Single Tree-Search SphereDecoder (STS-SD) Oneof thetwominima in (14) corresponds to theMAPhypothesis sMAPwhile the other corresponds to the counter hypothesis Thecomputation of LLR can be expressed by
119871 (119909119894119887
) =
1
1198730
(119889MAP119894119887
minus 119889MAP
) if 119909MAP119894119887
= +1
1
1198730
(119889MAP
minus 119889MAP119894119887
) if 119909MAP119894119887
= minus1
(20)
with
119889MAP
=10038171003817100381710038171003817y minus RsMAP10038171003817100381710038171003817
2
minus 1198730119875 (sMAP
)
119889MAP119894119887
= min119904isin120594
MAP119894119887
1003817100381710038171003817y minus Rs1003817100381710038171003817
2
minus 1198730119875 (s)
sMAP= arg min
119904isin2119876sdot119872
1003817100381710038171003817y minus Rs1003817100381710038171003817
2
minus 1198730119875 (s)
(21)
where 120594MAP119894119887
denotes the bitwise counter hypothesis of theMAP hypothesis which is obtained by searching over all thesolutions with the 119887th bit of the 119894th symbol opposite to thecurrentMAPhypothesisOriginally theMAPhypothesis andthe counter hypotheses can be found through repeating thetree search [39] The repeated tree search yields a large com-putational complexity cost To overcome this the single tree-search algorithm [13 40] was developed to compute all theLLRs concurrently The 119889
MAP metric and the corresponding119889MAP119894119887
metrics are updated through one tree-search processThrough the use of extrinsic LLR clipping method the STS-SD algorithm can be tunable between the MAP performanceand hard-output performance The implementations of STS-SD have been reported in [14 15]
423 SISO K-Best Decoder K-Best algorithm is a breadth-first search based algorithm in which the tree is traversedonly in the forward direction [41] This approach searchesonly a fixed number 119870 of paths with best metrics at eachdetection layer Figure 4(b) shows an example of the treesearch with 119873
119905= 2 The algorithm starts by extending the
root node to all possible candidates It then sorts the newpaths according to their metrics and retains the 119870 paths withsmallest metrics for the next detection layer
K-Best algorithm is able to achieve near-optimal perfor-mance with a fixed and affordable complexity for parallelimplementation Yet the major drawbacks ofK-Best decoderare the expansion and the sorting operations that are verytime consuming Several proposals have been drawn in theliterature to approximate the sorting operations such asrelaxed sorting [42] and distributed sorting [43] or evento avoid sorting using on-demand expansion scheme [44]Moreover similarly as LSD K-Best decoder suffers frommissing counter-hypothesis problem due to the limited listsize Numerous approaches have been proposed to addressthis problem such as smart candidates adding [45] bit flip-ping [46] and path augmentation and LLR clipping [12 35]
43 Interference-Cancellation-Based Detection Interference-cancellation-based detection can be carried out either in aparallel way as inMMSE-IC [8 9] or in a successive way as inVBLAST [47]
431 Minimum Mean Square Error-Interference Cancella-tion (MMSE-IC) Equalizer MMSE-IC equalizer can be per-formed using two filters [4]The first filter p
119894is applied to the
received vector y and the second filter q119894is applied to the
estimated vector s in order to cancel the interference fromother layers The equalized symbol
119894can be written as
119894= p119867
119894y minus q119867
119894s119894
with 119894 isin [1 119873119905] (22)
where s119894denotes the estimated vector given by the
previous iteration with the 119894th symbol omitteds119894
= [1 sdot sdot sdot 119894minus1
0 119894+1
sdot sdot sdot 119873119905]
119894is calculated by the
soft mapper as 119894
= E[119904119894] = sum
119904isin2119876 119904119875 (119904
119894= 119904) [48] The filters
p119894and q
119894are optimized using the MMSE criterion and are
given in [6 24]For the first iteration since no a priori information is
available the equalization process is reduced to the classicalMMSE solution
119894= (H119867H +
1205902
119899
1205902119904
I119873119905
)
minus1
H119867y (23)
The equalized symbols 119894are associated with a bias factor 120573
119894
in addition to some residual noise plus interferences 120578119894
119894= 120573
119894119904119894+ 120578
119894 (24)
These equalized symbols are then used by the soft demapperto compute the LLR values using the max-log-MAP approxi-mation [48]
119871 (119909119894119887
) =1
1205902120578119894
(min119904119894isin120594minus1
119894119887
1003816100381610038161003816119894 minus 120573119894119904119894
1003816100381610038161003816
2
minus min119904119894isin120594+1
119894119887
1003816100381610038161003816119894 minus 120573119894119904119894
1003816100381610038161003816
2
) (25)
MMSE-IC equalizer requires 119873119905matrix inversions for each
symbol vector For this reason several approximations of
MMSE-IC were proposed For example in [9] a low-complexity approach of MMSE-IC is described by perform-ing a single matrix inversion without performance loss Thisalgorithm is referred to as LC-MMSE-IC
432 Successive Interference Cancellation (SIC) EqualizerThe SIC-based detector was initially used in the VBLASTsystems In VBLAST architecture [47] a successive cancel-lation step followed by an interference nulling step is usedto detect the transmitted symbols However this methodsuffers from error propagation An improved VBLAST foriterative detection and decoding is described in [49] At thefirst iteration an enhanced VBLAST which takes decisionerrors into account is employed [24] When the a prioriLLRs are available from the channel decoder soft symbols arecomputed by a soft mapper and are used in the interferencecancellation To describe the enhanced VBLAST algorithmwe assume that the detection order has been made accordingto the optimal detection order [47] For the 119894th step thepredetected symbol vector s
119894minus1until step 119894 minus 1 is canceled out
from the received signal
y119894= y minus H
1119894minus1s119894minus1
(26)
where s119894minus1
= [1
2
sdot sdot sdot 119894minus1
] and H1119894minus1
=
[h1h2
sdot sdot sdot h119894minus1
] with h119894being the 119894th column of H
Then the estimated symbol 119894is obtained using a filtered
matrix W119894based on the MMSE criterion that takes decision
errors into account [11 49]
119894= W119867
119894y119894= 120573
119894119904119894+ 120578
119894
W119894= 120590
2
119904(HΣ
119894H119867 + 119873
0I119873119903
)minus1
h119894
(27)
Σ119894is the decision error covariance matrix defined as
Σ119894=
119894minus1
sum
119895=1
1205982
119895e119895e119879119895
+
119873119905minus119894+1
sum
119895=119894
1205902
119904e119895e119879119895
1205982
119895= E
10038161003816100381610038161003816119904119895
minus 119895
10038161003816100381610038161003816
2
| 119895minus1
(28)
8 Mobile Information Systems
where e119894denotes a unit vector having zero components except
the 119894th component which is oneA soft demapper is then used to compute LLRs according
to (25) We refer to this algorithm as improved VBLAST (I-VBLAST) in the sequel
44 Low-Complexity K-Best Decoder The low-complexityK-Best (LC-K-Best) decoder recently proposed in [20] usestwo improvements over the classical K-Best decoder for thesake of lower complexity and latency The first improvementsimplifies the hybrid enumeration of the constellation pointsin real-valued systemmodel when the a priori information isincorporated into the tree search using two look-up tablesThe second improvement is to use a relaxed on-demandexpansion that reduces the need of exhaustive expansionand sorting operations The LC-K-Best algorithm can bedescribed as follows
The preprocessing step is as follows
(1) Input yH 119870CalculateH = QR y = Q119867y
(2) Enumerate the constellation symbols based on119898119860 for
all layers
The tree-search step is as follows
(1) Set layer to 2119873119905 listL = 0 119889
2119873119905+1
= 0
(a) expand all radic2119876 possible constellation nodes(b) calculate the corresponding PEDs(c) if radic2119876 gt 119870 select the 119870 best nodes and store
them in the listL2119873119905
(2) For layer 119894 = 2119873119905minus 1 1
(a) enumerate the constellation point according to119898119862
119894of the 119870 surviving paths in the listL
119894+1
(b) find the first child (FC) based on 119898119862
119894and 119898
119860
119894for
each parent nodes(c) compute their PEDs(d) select 119860 best children with smallest PEDs
among the 119870 FCs and add them to the listL119894
(e) if |L119894| lt 119870 find the next child (NC) of the
selected parent nodesCalculate their PEDs and go to step (2)(d)
(f) else move to the next layer 119894 = 119894 minus 1 and go tostep (2)
(3) If 119894 = 1 calculate the LLR as in (19)In the case of missing counter hypothesis LLR clip-ping method is used
It has been shown in [20] that the LC-K-Best decoderachieves almost the same performance as the classical K-Best decoder with different modulations Moreover thecomputational complexity in terms of the number of visitednodes is significantly reduced specially in the case of high-order modulations
5 Convergence of IterativeDetection-Decoding
The EXtrinsic Information Transfer (EXIT) chart is a usefultool to study the convergence behavior of iterative decod-ing systems [50] It describes the exchange of the mutualinformation in the iterative process in order to predict therequired number of iterations the convergence threshold(corresponding to the start of the waterfall region) and theaverage decoding trajectory
In the iterative receiver considered in our study two itera-tive processes are performed one inside the channel decoder(turbo or LDPC) and the other between the MIMO detectorand the channel decoder For simplicity we separately studythe convergence of the channel decoding and the MIMOdetection We denote by 119868
1198601and 119868
1198602the a priorimutual input
information of the MIMO detector and the channel decoderrespectively and by 119868
1198641and 119868
1198642their corresponding extrinsic
mutual output informationThe mutual information 119868
119909(119868119860or 119868
119864) can be computed
through Monte Carlo simulation using the probability den-sity function 119901
119871119909
[50]
119868119909
=1
2sum
119909isinminus11
int
+infin
minusinfin
119901119871119909
(119871119909
| 119909)
sdot log2
2119901119871119909
(119871119909
| 119909)
119901119871119909
(119871119909
| minus1) 119901119871119909
(119871119909
| +1)119889119871
119909
(29)
A simple approximation of the mutual information is used inour analysis [51]
119868119909
asymp 1 minus1
119871119887
119871119887minus1
sum
119899=0
log2
(1 + exp (minus119909119871119909)) (30)
where 119871119887is the number of transmitted bits and 119871
119909is the LLR
associated with the bit 119909 isin minus1 +1The a priori information 119871
119860can be modeled by applying
an independentGaussian randomvariable 119899119860with zeromean
and variance 1205902
119860in conjunction with the known transmitted
information bits 119909 [50]
119871119860
= 120583119860
119909 + 119899119860
where 120583119860
=1205902
119860
2 (31)
For each given mutual information value 119868119860
isin [0 1] 1205902
119860can
be computed using the following equation [52]
120590119860
asymp (minus1
1198671
log2
(1 minus 11986811198673
119860))
121198672
(32)
where 1198671
= 03073 1198672
= 08935 and 1198673
= 11064At the beginning the a priori mutual information is as
follows 1198681198601
= 0 and 1198681198602
= 0 Then the extrinsic mutualinformation 119868
1198641of the MIMO detector becomes the a priori
mutual information 1198681198602
of the channel decoder and so onand so forth (ie 119868
1198641= 119868
1198602and 119868
1198642= 119868
1198601) For a successful
decoding there must be an open tunnel between the curves
Mobile Information Systems 9
Table 1 Simulation parameters for convergence analysis
MIMO system 4 times 4 spatial multiplexingModulation 16-QAM Gray mappingChannel Flat Rayleigh fading
DetectorSTS-SD 119871 clip = plusmn8
LC-119870-Best 119870 = 16 119871 clip = plusmn3
MMSE-IC
Channel decoder
LTE turbo code [13 15]119900
Block length 119870119887
= 1024 119877119888
= 12
LDPC code (IEEE 80211n)Codeword length 119873
119887= 1944 119877
119888= 12
Interleaver Random size = 1024 (turbo code case)
the exchange of extrinsic information can be visualized as aldquozigzagrdquo decoding trajectory in the EXIT chart
To visualize the exchange of extrinsic information of theiterative receiver we present the MIMO detector and thechannel decoder characteristics into a single chart For ourconvergence analysis a 4 times 4 MIMO system with 16-QAMconstellation turbo decoder and LDPCdecoder (119877
119888= 12) is
considered Table 1 summarizes the main parameters for theconvergence analysis
Figure 5 shows the extrinsic information transfer char-acteristics of MIMO detectors at different 119864
119887119873
0values
As the I-VBLAST detector performs successive interferencecancellation at the first iteration and parallel interference can-cellation of the soft estimated symbols for the rest iterationsit is less intuitive to present its convergence in the EXITchart Therefore the convergence analysis of VBLAST is notconsidered
It is obvious that the characteristics of the detectorsare shifted upward with the increase of 119864
119887119873
0 We show
that for low 119864119887119873
0(0 dB) and for low mutual information
(lt01) MMSE-IC performs better than LC-K-Best decoderHowever with larger mutual information its performance islower Moreover the mutual information of STS-SD is higherthan LC-K-Best decoder and MMSE-IC for different 119864
119887119873
0
For higher 119864119887119873
0(5 dB) MMSE-IC presents lower mutual
information than other decoders when 1198681198601
lt 09Figure 6 shows the EXIT chart for 119864
119887119873
0= 2 dB with
several MIMO detectors namely STS-SD LC-K-Best andMMSE-IC We note that the characteristic of the channeldecoder is independent of 119864
119887119873
0values It is obvious that
the extrinsic mutual information of the channel decoderincreases with the number of iterations We see that after 6 to8 iterations in the case of turbo decoder (Figure 6(a)) thereis no significant improvement on the mutual informationMeanwhile in the case of LDPC decoder in Figure 6(b) 20iterations are enough for LDPC decoder to converge
By comparing the characteristics of STS-SD LC-K-Bestdecoder andMMSE-IC equalizer with both coding schemeswe notice that STS-SD has a larger mutual informationat its output LC-K-Best decoder has slightly less mutualinformation than STS-SD MMSE-IC shows low mutualinformation levels at its output compared to other algorithms
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
STS-SDLC-K-Best
MMSE-IC
EbN0 = 0 2 5dB
5dB
2dB
0dBI E1I
A2
IA1 IE2
Figure 5 Extrinsic information transfer characteristics of MIMOdetectors (STS-SD LC-K-Best andMMSE-IC) at119864
119887119873
0= 0 2 5 dB
in a 4 times 4 MIMO system using 16-QAM
when 1198681198601
lt 09 while for 1198681198601
gt 09 the extrinsic mutualinformation is comparable to others
In the case of turbo decoder (Figure 6(a)) with 119868in =
8 3 outer iterations are sufficient for STS-SD to convergeat 119864
119887119873
0= 2 dB However the same performance can
be attained by performing 4 outer iterations with only 2inner iterations Similarly LC-K-Best decoder shows anequivalent performance but slightly higher119864
119887119873
0is required
The convergence speed of LC-K-Best decoder is a bit lowerthan STS-SD which requires more iterations to get the sameperformance The reason is mainly due to the unreliabilityof LLRs caused by the small list size (L = 16) In thecase of MMSE-IC the characteristic presents a lower mutualinformation than the LC-K-Best decoder when 119868
1198601lt 09
Therefore an equivalent performance can be obtained athigher 119864
119887119873
0or by performing more iterations
In a similar way we study the convergence of MIMOdetection algorithms with LDPC decoder Figure 6(b) showsthe EXIT chart at 119864
119887119873
0= 2 dBThe same conclusion can be
retrieved as in the case of turbo decoder We can see that at119864119887119873
0= 2 dB a clear tunnel is observed between the MIMO
detector and the channel decoder characteristics allowingiterations to bring improvement to the system SimilarlySTS-SD offers higher mutual information than LC-K-Bestdecoder andMMSE-IC equalizer which suggests its superiorsymbol detection performance
Additionally the average decoding trajectory resultingfrom the free-run iterative detection-decoding simulationsis illustrated in Figure 6 at 119864
119887119873
0= 2 dB with 119868out = 4
119868in = 2 and 119868out = 4 119868in = 5 in the case of turbo decoder andLDPC decoder respectively The decoding trajectory closely
10 Mobile Information Systems
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
TurboSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 2 6 8
(a)
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
LDPCSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 5 10 20
(b)
Figure 6 EXIT chart for MIMO detectors (STS-SD LC-K-Best and MMSE-IC) and channel decoder (a) LTE turbo decoder 119870119887
= 1024119877119888
= 12 and (b) LDPC decoder 119873119887
= 1944 119877119888
= 12 at 119864119887119873
0= 2 dB in a 4 times 4 MIMO system using 16-QAM
matches the characteristics in the case of STS-SD and LC-K-Best decoders The little difference from the characteristicsafter a few iterations is due to the correlation of extrinsicinformation caused by the limited interleaver depth In thecase of MMSE-IC the decoding trajectory diverges fromthe characteristics for high mutual information because theequalizer uses the a posteriori information to compute softsymbols instead of the extrinsic information
The best trade-off scheduling of the required numberof iterations is therefore 119868out iterations in the outer loopand a total of 8 iterations inside the turbo decoder and 20iterations inside the LDPC decoder distributed across these119868out iterations
6 Performance and Complexity Evaluation ofIterative Detection-Decoding
In this section we evaluate and compare the performance andthe complexity of differentMIMOdetectors namely STS-SDLC-K-Best decoder MMSE-IC and I-VBLAST equalizerswith different channel coding techniques (turbo LDPC)A detailed analysis of the performance and the complexitytrade-off ofMIMOdetection with LTE turbo decoder and 16-QAM modulation in a Rayleigh channel has been discussedin [24] Herein the performance and the complexity ofthe receiver with LDPC decoder are investigated Moreoverseveral modulations and coding schemes are consideredto quantify the gain achieved by such iterative receiver indifferent channel environments Consequently a comparative
study is conducted in iterative receiver with both codingschemes (turbo LDPC)
For the turbo code the rate 13 turbo encoder specifiedin 3GPP LTE with a block length 119870
119887= 1024 is used in the
simulations Puncturing is performed in the rate matchingmodule to achieve an arbitrary coding rate 119877
119888(eg 119877
119888= 12
34)Meanwhile the LDPC encoder specified in IEEE 80211nis consideredThe encoder is defined by a parity checkmatrixthat is formed out of square submatrices of sizes 27 54 or 81Herein the codeword length of size 119873
119887= 1944 with coding
rate of 119877119888
= 12 and 34 is considered
61 Performance Evaluation The simulations are first carriedout in Rayleigh fading channel to view general performanceof the iterative receivers Real channel models will be consid-ered to evaluate the performance in more realistic scenariosTherefore the 3GPP LTE-(A) channel environments withlow medium and large delay spread values and Dopplerfrequencies are considered The low spread channel is theExtended Pedestrian A (EPA) model which emulates theurban environment with small cell sizes (120591rms = 43 ns) Themedium spread channel (120591rms = 357 ns) is the ExtendedVehicular A (EVA) model The Extended Typical Urban(ETU) model is the large spread channel which has alarger excess delay (120591rms = 991 ns) and simulates extremeurban suburban and rural cases Table 2 summarizes thecharacteristic parameters of these channel environments Forall cases the channel is assumed to be perfectly known atthe receiver Table 3 gives the principle parameters of thesimulations The performance is measured in terms of bit
Mobile Information Systems 11
Table 2 Characteristic parameters of the investigated channelmodels
error rate (BER) with respect to signal-to-noise ratio (SNR)per bit 119864
119887119873
0
119864119887
1198730
=119864119904
1198730
+ 10 sdot log10
1
119877119888119876119873
119905
[dB] (33)
In our previous study [24] the performance of MIMOdetectors with LTE turbo decoder is evaluated in a Rayleighchannel with various outer and inner iterations It has beenshown that the performance is improved by about 15 dB at aBER level of 1 times 10
minus5 with 4 outer iterations It has been alsoshown that no significant improvement can be achieved after4 outer iterations this improvement is less than 02 dB with 8outer iterations
Similarly to turbo decoder we fix the number of inneriterations inside LDPC decoder to 20 while varying thenumber of outer iterations Figure 7 shows BER performanceof MIMO detectors with LDPC decoder in Rayleigh channelwith 119868in = 20 iterations and 119868out = 1 2 4 or 8 iterationsSTS-SD is used with a LLR clipping level of 8 which givesclose toMAPperformancewith considerable reduction in thecomplexity For LC-K-Best decoder a LLR clipping level of 3is used in the case of missing counter hypothesis We show
0 1 2 3 4 5
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
Iout = 4
Iout = 8
Iout = 2
Iout = 1
EbN0 (dB)minus1
Figure 7 BER performance of a 4 times 4 spatial multiplexing systemwith 16-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) in a Rayleigh fading channel 119868out = 1 2 4 8 119868in =
20 LDPC decoder 119873119887
= 1944 and 119877119888
= 12
that performance improvement of 15 dB is observed with 4outer iterations For 119868out = 8 iterations the improvementis less than 02 dB Therefore 119868out = 4 iterations will beconsidered in the sequel
Figure 8 shows the BER performance of 4 times 4 16-QAMsystem in a Rayleigh fading channel with 119868out = 4 and 119868in = 2
in the case of turbo decoder and 119868in = [3 4 6 7] in the caseof LDPC decoder The notation 119868in = [3 4 6 7] denotes that3 inner iterations are performed in the 1st outer iteration 4inner iterations in the 2nd outer iteration and so on Theperformances of STS-SD with 119868in = 8 and 119868in = 20 foreach outer iteration in the case of turbo decoder and LDPCdecoder are also plotted as a reference In the case of turbodecoder we show that performing 119868in = 8 and 119868out = 4
iterations does not bring significant improvement on theperformance compared to the case when 119868in = 2 and 119868out =
4 iterations are performed Similarly in the case of LDPCdecoder the performance of 119868in = 20 and 119868out = 4 iterationsis comparable to the performance of 119868in = [3 4 6 7] and119868out = 4 iterations Hence using a large number of iterationsdoes not seem to be efficientwhich proves the results obtainedin the convergence analysis of Section 5
By comparing the algorithms LC-K-Best decoder showsa degradation of less than 02 dB compared to STS-SD ata BER level of 2 times 10
minus5 However it outperforms MMSE-IC and I-VBLAST equalizer by about 02 dB at a BERlevel of 2 times 10
minus5 MMSE-IC and I-VBLAST show almostthe same performance In addition in the case of LDPCdecoder (Figure 8(b)) we notice that increasing the numberof inner iterations for each outer iteration 119868in = [3 4 6 7]
12 Mobile Information Systems
0 05 1 15 2 25 3
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 8
(a) LTE turbo decoder
BER
0 05 1 15 2 25 3
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 20
STS-SD Iin = 5
STS-SD
LC-K-BestI-VBLASTMMSE-IC
(b) LDPC decoder
Figure 8 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
shows slightly better performance than performing an equalnumber of iterations 119868in = 5 for each outer iteration
In the case of high-order modulation higher spectralefficiency can be achieved at a cost of increased symboldetection difficulty Figure 9 shows the BER performance of64-QAM with 119877
119888= 34 We see that LC-K-Best decoder
with a list size of 32 presents the similar performance as STS-SD However I-VLAST equalizer and MMSE-IC equalizerpresent degradation of more than 2 dB at a BER level of 1 times
10minus5 compared to LC-K-Best decoder Therefore LC-K-Best
decoder is more robust in the case of high-ordermodulationsand high coding rates The figure also shows that the BERperformance of LDPC decoder is almost identical to that ofthe turbo decoder
In order to summarize the performance of differentdetectors with different channel decoders we provide the119864119887119873
0values achieving a BER level of 1 times 10
minus5 in Table 4The values given in the parentheses of the table represent theperformance loss compared to STS-SD
Next we evaluate the performance of the iterative receiverin more realistic channel environments Figures 10 11 and 12show the BER performance of the detectors with the channeldecoders on EPA EVA and ETU channels receptivelySimilar behaviors can be observed with LTE turbo decoderand with LDPC decoder
In EPA channel (Figure 10) we see that LC-K-Bestdecoder achieves similar performance as STS-SD in the caseof 64-AQM and presents a degradation less than 02 dB in the
case of 16-QAM Meanwhile MMSE-IC presents significantperformance loss of more than 6 dB in the case of 64-QAMand 119877
119888= 34 With 16-QAM and 119877
119888= 12 the degradation
of MMSE-IC compared to LC-K-Best decoder is about 1 dBat a BER level of 1 times 10
minus4In EVA channel (Figure 11) the performance loss of
MMSE-IC compared to LC-K-Best decoder is reduced toapproximately 5 dB with 64-QAM and 05 dB with 16-QAMLC-K-Best decoder presents a degradation of about 01sim03 dB compared to STS-SD in the case of 16-QAM and 64-QAM
Similarly in ETU channel (Figure 12) MMSE-IC presentsa performance degradation compared to LC-K-Best decoderThis degradation is less than 4 dB in the case of 64-QAM andless than 05 dB in the case of 16-QAM We notice also thatthe LC-K-Best decoder is comparable to STS-SD in the caseof 64-QAM and has a degradation of 02 dB in the case of 16-QAM
Comparing the performance of the iterative receiver indifferent channels it can be seen that iterative receiverspresent the best performance in ETU channel compared toEPA and EVA channels This is due to the high diversity ofETU channel At a BER level of 1 times 10
minus4 the performancegain in ETU channel in the case of LTE turbo decoder is about08 dB 13 dB compared to EVA channel with 16-QAM and64-QAM respectively In the case of LDPC decoder this gainis 04 dB and 1 dB with 16-QAM and 64-QAM respectivelyHowever in EPA channel the performance gain in ETU
Mobile Information Systems 13
3 4 5 6 7 8 9 10 11 12 13
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
EbN0 (dB)
(a) LTE turbo decoder
3 4 5 6 7 8 9 10 11 12
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)
STS-SD Iin = 20
STS-SDLC-K-Best
I-VBLASTMMSE-IC
(b) LDPC decoder
Figure 9 BER performance of a 4 times 4 spatial multiplexing system with 64-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
0values achieving a BER level of 1times 10minus5 for different detectors and channel decoders (turbo LDPC) in 4times 4 spatialmultiplexing
system with 16-QAM 119877119888
= 12 and 64-QAM 119877119888
= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
STS-SD 25 dB 102 dB 27 dB 96 dBLC-119870-Best 26 dB (minus01) 102 dB (00) 28 dB (minus01) 96 dB (00)I-VBLAST 28 dB (minus03) 128 dB (minus26) 29 dB (minus02) 12 dB (minus24)MMSE-IC 28 dB (minus03) 128 dB (minus26) 30 dB (minus03) 12 dB (minus24)lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
channel in the case of turbo decoder or LDPC decoder ismore than 1 dB with 16-QAM and 64-QAM
Table 5 summarizes the 119864119887119873
0values achieving a BER
level of 1times10minus4 of different detectors combined with different
channel decoders andmodulation orders in different channelmodels The values given in the parentheses in the tablerepresent the performance loss compared to STS-SD Asindicated in Table 5 the iterative receivers with turbo decoderand LDPC decoder have comparable performance with acoding rate 119877
119888= 12 (16-QAM) However with 119877
119888= 34
(64-QAM) the receivers with LDPC decoder present slightlya better performance especially in ETU channel (06 dB)
From these results we show that the iterative receiversubstantially improves the performance of coded MIMOsystems either with turbo decoder or with LDPC decoderin Rayleigh channel (Figures 8 and 9) and in more realisticchannels (Figures 10 11 and 12) Moreover we show that
performing a large number of inner iterations does notbring significant improvement In addition we show thatthe LC-K-Best decoder achieves a good performance withdifferent modulations and channel coding schemes Thefigures suggest that the BER performance of the iterativereceiver with turbo decoder is almost comparable to that ofthe LDPC decoder It is therefore meaningful to evaluate thecomputational complexity of the iterative receivers with bothdecoding techniques as it will be discussed in the next section
62 Complexity Evaluation The computational complexityhas significant impact on the latency throughput and powerconsumption of the deviceTherefore the receiver algorithmsshould be optimized to achieve a good trade-off betweenperformance and cost In this part we evaluate the com-putational complexity of the iterative receiver in terms ofbasic operations such as addition subtractionmultiplication
14 Mobile Information Systems
1 3 5 7 9 11 13 15 17 19
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
(a) EPA turbo 119868out = 4 119868in = 2
BER
1 3 5 7 9 11 13 15 17 19
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12 64-QAM
Rc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EPA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 10 BER performance of a 4 times 4 spatial multiplexing with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) on EPA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944 119868out = 4
119868in = [3 4 6 7]
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
STS-SDLC-K-Best
MMSE-IC
16-QAMRc = 12
64-QAMRc = 34
(a) EVA turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EVA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 11 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Best and MMSE-IC) on EVA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944
and 119868out = 4 119868in = [3 4 6 7]
Mobile Information Systems 15
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(a) ETU turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) ETU LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 12 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SDLC-K-Best and MMSE-IC) on ETU channel model (a) LTE turbo decoder 119870
minus4 in LTE channel models for different detectors and channel decoders (turbo LDPC)in 4 times 4 spatial multiplexing system with 16-QAM 119877
119888= 12 and 64-QAM 119877
119888= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
EPASTS-SD 63 dB 140 dB 62 dB 138 dB
LC-119870-Best 65 dB (minus02) 139 dB (+01) 64 dB (minus02) 139 dB (minus01)MMSE-IC 74 dB (minus11) gt20 dB (gtminus6) 80 dB (minus18) gt20 dB (gtminus6)
EVASTS-SD 52 dB 143 dB 53 dB 134 dB
LC-119870-Best 54 dB (minus02) 144 dB (minus01) 56 dB (minus03) 137 dB (minus03)MMSE-IC 58 dB (minus06) 190 dB (minus47) 60 dB (minus07) 185 dB (minus51)
ETUSTS-SD 44 dB 130 dB 49 dB 124 dB
LC-119870-Best 46 dB (minus02) 130 dB (00) 49 dB (00) 124 dB (00)MMSE-IC 49 dB (minus05) 175 dB (minus35) 51 dB (minus02) 16 dB (minus36)
lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
division square root extraction maximization and look-up table check (which are denoted by ADD SUB MULDIV SQRT Max and LUT resp) To this end the completecomparison of the iterative receivers with both channeldecoders (turbo LDPC) and with several modulations andcoding rates is carried out
621 Complexity of Iterative Receiver The complexity ofiterative receiver depends on theMIMOdetector the channeldecoder and the number of innerouter iterationsThis com-plexity can be expressed by
sdot [119862det1 + (119868out minus 1) sdot 119862deti] (34)
where 119862det1 denotes the complexity of the first iterationof MIMO detection algorithm per symbol vector withouttaking into consideration the a priori information 119862detidenotes the complexity per iteration per symbol vector takinginto consideration the a priori information 119862dec denotesthe complexity of the channel decoder per iteration perinformation bit 119873bit is the number of information bit at theinput of the encoder 119873symb is the number of symbol vectors119873symb and 119873bit are linked by the following relation
119873symb =119873bit
119876119877119888119873119905
= 120572119873bit with 120572 =1
119876119877119888119873119905
(35)
where 119876 is the number of bits in the constellation symbol 119877119888
is the coding rate and119873119905is the number of transmit antennas
16 Mobile Information Systems
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
MIMO detectors have been also widely investigated such asthe minimum mean square error-interference cancellation(MMSE-IC) [8 9] improved VBLAST (I-VBLAST) [10 11]list sphere decoder (LSD) [12] single tree-search spheredecoder (STS-SD) [13ndash15] K-Best decoder [16ndash20] and fixedsphere decoder (FSD) [21ndash23] Among them MMSE-IC andI-VBLAST present low computational complexity but theyare not able to fully exploit the spatial diversity of MIMOsystem Meanwhile the sphere decoder is able to achievesuperior performance However the sphere decoder uses adepth-first searchmethodTherefore its computational com-plexity varies significantly with respect to the channel con-dition yielding prohibitive worst-case complexity Moreoverthe sphere decoder suffers from variable throughput due toits sequential tree-search strategy which makes it unsuitablefor parallel implementation In contrast the breadth-firstsearch based K-Best and FSD algorithms are hence moreattractive for practical implementation than sphere decodingas they can offer stable throughput at a cost of acceptableperformance loss
Despite these efforts it is still very challenging to developa high speed iterative MIMO receiver to meet the highthroughput requirements of future wireless communicationsystems at affordable complexity and implementation costIn [24] the performance-complexity trade-offs of iterativeMIMO receiver have been investigated However the investi-gation is limited to the turbo channel coding and theoreticalchannel cases In this contribution the performance andthe complexity of iterative MIMO receiver are evaluated ina much broader and more practical scope We investigatein depth the soft joint iterative detection schemes withvarious symbol detection schemes various soft-input soft-output channel decoders and various ways of constructingjoint loops under different channel conditions In particularthe most representative modern channel coding schemesincluding LTE turbo code and LDPC code are consideredSeveral LTE multipath channel models are employed in thesimulation to evaluate the performance in real propagationscenarios Consequently a detailed comparative study is con-ducted among iterative receivers with different modulationsand channel coding schemes (turbo LDPC) It has beendemonstrated through the comparison that LC-K-Best basedreceiver achieves a best trade-off between performance andcomplexity among the iterative MIMO receivers consideredin this work
The remainder of this paper is organized as followsSection 2 presents the MIMO-OFDM system model andthe concept of iterative detection-decoding process Channeldecoding based on turbo decoder and LDPC decoder isdescribed in Section 3 Section 4 briefly reviews the mostrelevant SISO MIMO detection algorithms based on spheredecoder LC-K-Best decoder and interference canceller InSection 5 the convergence behavior of the iterative receiversis discussed using extrinsic information transfer (EXIT) chartto retrieve to required number of inner and outer itera-tions Section 6 illustrates the performance of our proposedapproaches in LTE-based channel environments Then thecomputational complexity of the receivers with both turboand LDPC coding techniques is evaluated and compared
with different modulation orders and coding rates Section 7concludes the paper
2 System Model
21 MIMO-OFDM System Model We consider a MIMO-OFDM system based on bit-interleaved coded modulation(BICM) scheme [25] with 119873
119905transmit antennas and 119873
119903
receive antennas (119873119903
ge 119873119905) as depicted in Figure 1
At the transmitter the information bits of length 119870119887are
first encoded by a channel encoder which outputs a codewordc of length 119873
119887with a coding rate 119877
119888= 119870
119887119873
119887 The channel
encoder can be a turbo encoder or an LDPC encoder Theencoded bits are then randomly interleaved andmapped intocomplex symbols of 2
119876 quadrature amplitude modulation(QAM) constellation where 119876 is the number of bits persymbol The symbols are mapped into 119873
119905transmit antennas
using either space-time block coding (STBC) schemes orspatial multiplexing (SM) schemes offering different diversitygain and multiplexing gain trade-offs Herein the SM-basedMIMO system is considered without loss of generality IFFTis applied to 119873
119888parallel symbols to obtain the time domain
OFDM symbols where 119873119888is the number of useful subcarri-
ers The symbols are then sent though the radio channel afterthe addition of the cyclic prefix (CP) which is assumed largerthan the maximum delay spread of the channel The timedomain symbol transmitted by the 119894th antenna is expressedas
119904119894(119899) =
1
radic119873FFT
119873FFTminus1
sum
119896=0
119878119894(119896) 119890
1198952120587119896119899119873FFT
minus119873119892
le 119899 le 119873FFT minus 1
(1)
where 119878119894(119896) is the symbol in the frequency domain before
IFFT 119873FFT is the size of the FFT and 119873119892is the length of
the CP The transmit power is normalized so that Ess119867 =
119864119904119873
119905I119873119905
where I119873119905
is the 119873119905
times 119873119905identity matrix The
transmission information rate is 119877119888
sdot 119873119905
sdot 119876 bits per channeluse
Using the OFDM technique the frequency-selective fad-ing channel is divided into a series of orthogonal and flat-fading subchannels The signal equalization is performed bya simple one-tap equalizer at the receiver Therefore after theremoval of CP FFT is performed to get the frequency domainsignal vector y
119896= [119910
1 1199102 119910
119873119903
]119879 that can be expressed as
y119896
= H119896s119896
+ n119896 (2)
where 119896 = 1 119873119888is the index of subcarriers For simplicity
the subcarrier index 119896 is omitted in the sequel H is the119873119903
times 119873119905channel matrix with its (119894 119895)th element ℎ
119894119895 the
channel frequency response of the channel link from 119895thtransmit antenna to 119894th receive antenna The coefficients ofthe channel matrix H are assumed to be perfectly known atthe receiver n = [119899
1 1198992 119899
119873119903
]119879 is the independent and
identically distributed (iid) additive white Gaussian noise(AWGN) vector with zero mean and variance of 119873
0= 120590
2
119899
Mobile Information Systems 3
Receiver
Transmitter Mapping
QAM MIMOmapper
Channel encoder
Channel decoder
DemapperQAM
MIMOdetector
Softmapper
IFFT
IFFT
FFT
FFT
u c e x s
H
y
LA1
LE1
Iout
LA2
LE2
Iin
Π
Π
Πminus1u xc
Figure 1 Block diagram of MIMO-OFDM system using bit-interleaved coded modulation with iterative detection and decoding
22 Iterative Detection-Decoding Principle At the receiver torecover the transmitted signal from interferences an iterativedetection-decoding process based on the turbo principle isapplied as depicted in Figure 1 The MIMO detector andthe channel decoder exchange soft information that is loglikelihood ratio (LLR) in each iteration
The MIMO detector takes the received symbol vector yand the a priori information 119871
1198601of the coded bits from the
channel decoder and computes the extrinsic information1198711198641
The MIMO detection algorithm can be the MAP algorithmor other suboptimal algorithms like STS-SDK-Best decoderI-VBLAST or MMSE-IC The extrinsic information is dein-terleaved and becomes the a priori information 119871
1198602for the
channel decoderThe channel decoder computes the extrinsicinformation 119871
1198642that is reinterleaved and fed back to the
detector as the a priori information 1198711198601
The channel decoding is performed either by an LTEturbo decoder or by an LDPC decoder which exchanges softinformation between their component decoders as describedin the next section In our iterative process we denote thenumber of outer iterations between the MIMO detector andthe channel decoder by 119868out and the number of iterationswithin the turbo decoder or LDPC decoder by 119868in
For QAM the mapping process can be done inde-pendently for real and imaginary part The system modelexpressed in (2) can be converted into an equivalent real-valued model
[
Re (y)
Im (y)] = [
Re (H) minus Im (H)
Im (H) Re (H)] [
Re (s)Im (s)
]
+ [
Re (n)
Im (n)]
(3)
where Re(sdot) and Im(sdot) represent the real and imaginary partsof a complex number respectively Each QAM constellationpoint is treated as two PAM symbols and the matrix dimen-sion is doubled However as shown in [17] the real-valuedmodel is more efficient for the implementation of the sphere
decoder Hence it will be used as the systemmodel in case ofsphere decoding in the following sections
3 Soft-Input Soft-Output Channel Decoder
Channel coding is used to protect the useful informationfrom channel distortion and noise by introducing someredundancy The state-of-the-art channel coding schemessuch as the LDPC [26] and turbo codes [3] can effectivelyapproach the Shannon bound LDPC codes are nowadaysadopted in many standards including IEEE 80211 and DVB-T2 as they achieve very high throughput due to inherentparallelism of the decoding algorithm In the meantime theturbo codes are also adopted in LTE LTE-A (binary turbocodes) and WiMAX (double binary turbo codes) In thispaper LDPC codes and LTE turbo codes are considered
31 Turbo Decoder Initially proposed in 1993 [3] turbocodes have attracted great attention due to the capacity-approaching performance The turbo encoder is constitutedby a parallel concatenation of two recursive systematicconvolutional encoders separated by an interleaver Thefirst encoder processes the original data while the secondprocesses the interleaved version of dataThemain role of theinterleaver is to reduce the degree of correlation between theoutputs of the component encoders
In LTE system the recursive systematic encoders with8 states and [13 15]
119900polynomial generators are adopted A
quadratic polynomial permutation (QPP) interleaver is usedas a contention free interleaver and it is suitable for paralleldecoding of turbo codes as illustrated in Figure 2(a) Themother coding rate is 13 Coding rates other than themotherrate can be achieved by puncturing or repetition using the ratematching technique
The turbo decoding is performed by two SISO compo-nent decoders that exchange soft information of their datasubstreams Each component decoder takes systematic orinterleaved information the corresponding parity informa-tion and the a priori information from the other decoder to
4 Mobile Information Systems
LTE QPPinterleaver
D D D
DDD
uk (Info)
1st encoder
2nd encoder
sk
p1k
p2k
s998400k
⨁
⨁ ⨁
⨁
⨁
⨁
⨁
⨁
(a) Turbo encoder
DEC2
DEC1
Lc(in) SP
Lc(p1)
Lc(s)
Lc(p2)
Πi
La(u)
Πminus1i ΠiIin
Le(u)
Le(u)
Le(u)
La(u)
Le(p2)
Le(p1)
PSLc(out)
(b) Turbo decoder
Figure 2 Structure of LTE turbo code with 119877119888
= 13 (a) turboencoder and (b) turbo decoder
compute the extrinsic information as shown in Figure 2(b)Two families of decoding algorithms can be used soft-output Viterbi algorithms (SOVA) [27 28] and maximum aposteriori (MAP) algorithm [29] The MAP algorithm offerssuperior performance but suffers from high computationalcomplexity Two suboptimal algorithms namely log-MAPand max-log-MAP are practically used [30] Herein log-MAPalgorithm is considered by using the Jacobian logarithm[30]
where 119891119888(|119909 minus 119910|) is a correction function that can be
computed using a small look-up table (LUT)The decoder computes the branch metrics (120574) and the
forward (120572) and the backward (120573) metrics between two statesin the trellis as follows
120574119896
(119904119896minus1
119904119896) = 119901 (119904
119896 119910119896
| 119904119896minus1
)
120572119896
(119904119896) =
lowastmax119904119896minus1
(120572119896minus1
(119904119896minus1
) + 120574119896
(119904119896minus1
119904119896))
120573119896
(119904119896) =
lowastmax119904119896+1
(120573119896+1
(119904119896+1
) + 120574119896
(119904119896 119904119896+1
))
(5)
The a posteriori LLRs of the information bits are computed as
119871 (119906119896) =
lowastmax119906119896=0
(120572119896minus1
(119904119896minus1
) + 120574119896
(119904119896minus1
119904119896) + 120573
119896(119904119896))
minuslowastmax119906119896=1
(120572119896minus1
(119904119896minus1
) + 120574119896
(119904119896minus1
119904119896) + 120573
119896(119904119896))
(6)
The component decoders exchange only the extrinsic LLRwhich is defined by
119871119890(119906119896) = 119871 (119906
119896) minus 119871
119886(119906119896) minus 119871
119904(119906119896) (7)
where 119871119886(119906119896) and 119871
119904(119906119896) correspond to the a priori informa-
tion from the other decoder and the systematic informationbits respectively
32 LDPC Decoder LDPC codes belong to a class of linearerror correcting block codes first proposed by Gallager [26]Their main advantages lie in their capacity-approaching per-formance and their low-complexity parallel implementations[31] LDPC codes can be represented by a parity checkmatrix HLDPC or intuitively through Tanner graph [32]Tanner graphs are bipartite graphs containing two types ofnodes the check nodes and the variable nodes as illustratedin Figure 3(a) It consists of 119872 check nodes (CN) whichcorrespond to the number of parity bits (ie number of rowsof HLDPC) and 119873 variable nodes (VN) corresponding to thenumber of bits in a codeword (ie number of columns ofHLDPC)
The optimal maximum a posteriori decoding of LDPCcodes is infeasible from the practical implementation point ofviewAlternatively LDPCdecoding is done using themessagepassing or belief propagation algorithms which iterativelypass messages between check nodes and variable nodes asshown in Figure 3(b) The belief propagation is denotedas the sum-product decoding because probabilities can berepresented as LLRs which allow the calculation of messagesusing sum and product operations
Let 119871V119894119895
be themessage from variable node 119894 to check node119895 and 119871
119888119895119894
the message from check node 119895 to variable node119894 Let 119881
119895and 119862
119894denote the set of adjacent variable nodes
connected to the check node 119895 and the set of adjacent checknodes connected to the variable node 119894 respectively where0 le 119895 le 119872 and 0 le 119894 le 119873 For the first iteration theinput to the LDPC decoder is the LLRs 119871(119888
119894) of the codeword
c which are used as an initial value of the extrinsic variablenode messages that is 119871V
119894119895
= 119871(119888119894) For the 119896th iteration the
algorithm can be summarized as follows(1) Each check node 119895 computes the extrinsic message to
its neighboring variable node 119894
119871119888119895119894
= 2tanhminus1 ( prod
1198941015840isin119881119895119894
tanh(
119871V1198941015840119895
2)) (8)
(2) Each variable node 119894 updates its extrinsic informationto the check node 119895 in the next iteration
119871V119894119895
= 119871 (119888119894) + sum
1198951015840isin119862119894119895
1198711198881198951015840119894
(9)
Mobile Information Systems 5
n = 6m = 4
VN (n)
CN (m)
0 1 1 0 0 1
1 1 1 0 1 0
1 0 0 1 1 1
0 0 1 1 0 1
=HLDPC
(a)
1 2 3 5 64
1 2 3 4
VNi
CNj
L13
L12
L31
L32
L34
Lc12Lc13
Lc16Lc31
Lc34
Lc35
Lc36
(b)
Figure 3 LDPC decoder (a) matrix representation and Tanner graph and (b) iterative decoding between VNs and CNs
(3) The a posterioriLLRof each codeword bit is computedas
119871119901
(119888119894) = 119871 (119888
119894) + sum
1198951015840isin119862119894
1198711198881198951015840119894
(10)
The decoding algorithm alternates between check node proc-essing and variable node processing until a maximum num-ber of iterations are achieved or until the parity check con-dition is satisfied When the decoding process is terminatedthe decoder outputs the a posteriori LLR
4 Soft-Input Soft-Output MIMO Detection
The aim of MIMO detection is to recover the transmittedvector s from the received vector y The state-of-the-artMIMO detection algorithms have been presented in [24]These algorithms can be divided into two main familiesnamely the tree-search-based detection and the interference-cancellation-based detection In this section we brieflyreview the main existing SISO MIMO detection algorithmsuseful for the following sections
41 MaximumA Posteriori Probability (MAP) Detection TheMAP algorithm achieves the optimum performance throughthe use of an exhaustive search over all 2
119876sdot119873119905 possible symbol
combinations to compute the LLR of each bit The LLR of the119887th bit in the 119894th transmit symbol 119909
119894119887 is given by
119871 (119909119894119887
) = log119875 (119909
119894119887= +1 | y)
119875 (119909119894119887
= minus1 | y)
= logsumsisin120594+1
119894119887
119901 (y | s) 119875 (s)
sumsisin120594minus1119894119887
119901 (y | s) 119875 (s)
(11)
where 120594+1
119894119887and 120594
minus1
119894119887denote the sets of symbol vectors in
which the 119887th bit in the 119894th antenna is equal to +1 and minus1respectively 119901(y | s) is the conditioned probability densityfunction given by
119901 (y | s) =1
(1205871198730)119873119903
exp(minus1
1198730
1003817100381710038171003817y minus Hs10038171003817100381710038172
) (12)
119875(s) represents the a priori information provided by thechannel decoder in the form of a priori LLRs
119871119860
(119909119894119887
) = log119875 (119909
119894119887= +1)
119875 (119909119894119887
= minus1) forall119894 119887
119875 (s) =
119873119905
prod
119894=1
119875 (119904119894) =
119873119905
prod
119894=1
119876
prod
119887=1
119875 (119909119894119887
)
(13)
The max-log-MAP approximation is commonly used in theLLR calculation with lower complexity [12]
119871 (119909119894119887
) asymp1
1198730
min120594minus1
119894119887
1198891 minus
1
1198730
min120594+1
119894119887
1198891 (14)
1198891
=1003817100381710038171003817y minus Hs1003817100381710038171003817
2
minus 1198730
119873119905
sum
119894=1
119876
sum
119887=1
log119875 (119909119894119887
) (15)
where 1198891represents the Euclidean distance between the
received vector y and lattice pointsHsBased on the a posteriori LLRs 119871(119909
119894119887) and the a priori
LLRs 119871119860
(119909119894119887
) the detector computes the extrinsic LLRs119871119864(119909119894119887
) as
119871119864
(119909119894119887
) = 119871 (119909119894119887
) minus 119871119860
(119909119894119887
) (16)
The MAP algorithm is not feasible due to its exponentialcomplexity since 2
119876sdot119873119905 hypotheses have to be considered
within each minimum term and for each bit Thereforeseveral suboptimal MIMO detectors have been proposedwith reduced complexity as will be briefly discussed in thefollowing sections
42 Tree-Search-Based Detection The tree-search-baseddetection methods generally fall into two main categoriesnamely depth-first search like the sphere decoder andbreadth-first search like the K-Best decoder
421 List Sphere Decoder (LSD) The basic idea of the spheredecoder is to limit the search space of the MAP solution to ahypersphere of radius 119903
119904around the received vector Instead
of testing all the hypotheses of the transmitted signal onlythe lattice points that lie inside the hypersphere are testedreducing the computational complexity [33]
sSD = arg minsisin2119876119873119905
1003817100381710038171003817y minus Hs1003817100381710038171003817
2
le 1199032
119904 (17)
6 Mobile Information Systems
Using the QR decomposition in real-valued model thechannel matrix H can be decomposed into two matrixes Qand R (H = QR) where Q is 2119873
119903times 2119873
119905orthogonal matrix
(Q119867Q = I2119873119905
) and R is 2119873119905
times 2119873119905upper triangular matrix
with real-positive diagonal elements [34] Therefore the dis-tance in (17) can be computed as yminusHs2 = yminusRs2 wherey = Q119867y is the modified received symbol vector Exploitingthe triangular nature of R the Euclidean distance metric 119889
1
in (15) can be recursively evaluated through the accumulatedpartial Euclidean distance (PED) 119889
and the a priori-based partial metric at the 119894th level respec-tively
This process can be illustrated by a tree with 2119873119905
+ 1
levels as depicted in Figure 4(a) The tree search starts at theroot level with the first child node at level 2119873
119905 The partial
Euclidean distance 1198892119873119905
in (18) is then computed If 1198892119873119905
issmaller than the sphere radius 119903
119904 the search continues at level
2119873119905minus1 and steps down the tree until finding a valid leaf node
at level 1List sphere decoder is proposed to approximate the MAP
detector [12] It generates a list L sub 2119876119873119905 that includes the
best possible hypotheses The LLR values are then computedfrom this list as
119871 (119909119894119887
) =1
1198730
minLcap120594minus1119894119887
1198891 minus
1
1198730
minLcap120594+1119894119887
1198891 (19)
The main issue of LSD is the missing counter-hypothesisproblem depending on the list sizeThe use of limited list sizecauses inaccurate approximation of the LLR due to missingsome counter hypotheses where no entry can be found inthe list for a particular bit 119909
119894119887= +1 minus1 Several solutions
have been proposed to handle this issue LLR clipping is afrequently used solution which consists simply to set the LLRto a predefined maximum value [12 35]
Several methods can be considered to reduce the com-plexity of the sphere decoder such as Schnorr-Euchner(SE) enumeration [36] layer ordering technique [34] andchannel regularization [37] Layer ordering technique allowsthe selection of the most reliable symbols at a high layerusing the sorted QR (SQR) decomposition However channelregularization introduces a biasing factor in the metricswhich should be removed in LLR computation to avoidperformance degradation as discussed in [38] In the sequelthe SQR decomposition is considered in the preprocessingstep
422 Single Tree-Search SphereDecoder (STS-SD) Oneof thetwominima in (14) corresponds to theMAPhypothesis sMAPwhile the other corresponds to the counter hypothesis Thecomputation of LLR can be expressed by
119871 (119909119894119887
) =
1
1198730
(119889MAP119894119887
minus 119889MAP
) if 119909MAP119894119887
= +1
1
1198730
(119889MAP
minus 119889MAP119894119887
) if 119909MAP119894119887
= minus1
(20)
with
119889MAP
=10038171003817100381710038171003817y minus RsMAP10038171003817100381710038171003817
2
minus 1198730119875 (sMAP
)
119889MAP119894119887
= min119904isin120594
MAP119894119887
1003817100381710038171003817y minus Rs1003817100381710038171003817
2
minus 1198730119875 (s)
sMAP= arg min
119904isin2119876sdot119872
1003817100381710038171003817y minus Rs1003817100381710038171003817
2
minus 1198730119875 (s)
(21)
where 120594MAP119894119887
denotes the bitwise counter hypothesis of theMAP hypothesis which is obtained by searching over all thesolutions with the 119887th bit of the 119894th symbol opposite to thecurrentMAPhypothesisOriginally theMAPhypothesis andthe counter hypotheses can be found through repeating thetree search [39] The repeated tree search yields a large com-putational complexity cost To overcome this the single tree-search algorithm [13 40] was developed to compute all theLLRs concurrently The 119889
MAP metric and the corresponding119889MAP119894119887
metrics are updated through one tree-search processThrough the use of extrinsic LLR clipping method the STS-SD algorithm can be tunable between the MAP performanceand hard-output performance The implementations of STS-SD have been reported in [14 15]
423 SISO K-Best Decoder K-Best algorithm is a breadth-first search based algorithm in which the tree is traversedonly in the forward direction [41] This approach searchesonly a fixed number 119870 of paths with best metrics at eachdetection layer Figure 4(b) shows an example of the treesearch with 119873
119905= 2 The algorithm starts by extending the
root node to all possible candidates It then sorts the newpaths according to their metrics and retains the 119870 paths withsmallest metrics for the next detection layer
K-Best algorithm is able to achieve near-optimal perfor-mance with a fixed and affordable complexity for parallelimplementation Yet the major drawbacks ofK-Best decoderare the expansion and the sorting operations that are verytime consuming Several proposals have been drawn in theliterature to approximate the sorting operations such asrelaxed sorting [42] and distributed sorting [43] or evento avoid sorting using on-demand expansion scheme [44]Moreover similarly as LSD K-Best decoder suffers frommissing counter-hypothesis problem due to the limited listsize Numerous approaches have been proposed to addressthis problem such as smart candidates adding [45] bit flip-ping [46] and path augmentation and LLR clipping [12 35]
43 Interference-Cancellation-Based Detection Interference-cancellation-based detection can be carried out either in aparallel way as inMMSE-IC [8 9] or in a successive way as inVBLAST [47]
431 Minimum Mean Square Error-Interference Cancella-tion (MMSE-IC) Equalizer MMSE-IC equalizer can be per-formed using two filters [4]The first filter p
119894is applied to the
received vector y and the second filter q119894is applied to the
estimated vector s in order to cancel the interference fromother layers The equalized symbol
119894can be written as
119894= p119867
119894y minus q119867
119894s119894
with 119894 isin [1 119873119905] (22)
where s119894denotes the estimated vector given by the
previous iteration with the 119894th symbol omitteds119894
= [1 sdot sdot sdot 119894minus1
0 119894+1
sdot sdot sdot 119873119905]
119894is calculated by the
soft mapper as 119894
= E[119904119894] = sum
119904isin2119876 119904119875 (119904
119894= 119904) [48] The filters
p119894and q
119894are optimized using the MMSE criterion and are
given in [6 24]For the first iteration since no a priori information is
available the equalization process is reduced to the classicalMMSE solution
119894= (H119867H +
1205902
119899
1205902119904
I119873119905
)
minus1
H119867y (23)
The equalized symbols 119894are associated with a bias factor 120573
119894
in addition to some residual noise plus interferences 120578119894
119894= 120573
119894119904119894+ 120578
119894 (24)
These equalized symbols are then used by the soft demapperto compute the LLR values using the max-log-MAP approxi-mation [48]
119871 (119909119894119887
) =1
1205902120578119894
(min119904119894isin120594minus1
119894119887
1003816100381610038161003816119894 minus 120573119894119904119894
1003816100381610038161003816
2
minus min119904119894isin120594+1
119894119887
1003816100381610038161003816119894 minus 120573119894119904119894
1003816100381610038161003816
2
) (25)
MMSE-IC equalizer requires 119873119905matrix inversions for each
symbol vector For this reason several approximations of
MMSE-IC were proposed For example in [9] a low-complexity approach of MMSE-IC is described by perform-ing a single matrix inversion without performance loss Thisalgorithm is referred to as LC-MMSE-IC
432 Successive Interference Cancellation (SIC) EqualizerThe SIC-based detector was initially used in the VBLASTsystems In VBLAST architecture [47] a successive cancel-lation step followed by an interference nulling step is usedto detect the transmitted symbols However this methodsuffers from error propagation An improved VBLAST foriterative detection and decoding is described in [49] At thefirst iteration an enhanced VBLAST which takes decisionerrors into account is employed [24] When the a prioriLLRs are available from the channel decoder soft symbols arecomputed by a soft mapper and are used in the interferencecancellation To describe the enhanced VBLAST algorithmwe assume that the detection order has been made accordingto the optimal detection order [47] For the 119894th step thepredetected symbol vector s
119894minus1until step 119894 minus 1 is canceled out
from the received signal
y119894= y minus H
1119894minus1s119894minus1
(26)
where s119894minus1
= [1
2
sdot sdot sdot 119894minus1
] and H1119894minus1
=
[h1h2
sdot sdot sdot h119894minus1
] with h119894being the 119894th column of H
Then the estimated symbol 119894is obtained using a filtered
matrix W119894based on the MMSE criterion that takes decision
errors into account [11 49]
119894= W119867
119894y119894= 120573
119894119904119894+ 120578
119894
W119894= 120590
2
119904(HΣ
119894H119867 + 119873
0I119873119903
)minus1
h119894
(27)
Σ119894is the decision error covariance matrix defined as
Σ119894=
119894minus1
sum
119895=1
1205982
119895e119895e119879119895
+
119873119905minus119894+1
sum
119895=119894
1205902
119904e119895e119879119895
1205982
119895= E
10038161003816100381610038161003816119904119895
minus 119895
10038161003816100381610038161003816
2
| 119895minus1
(28)
8 Mobile Information Systems
where e119894denotes a unit vector having zero components except
the 119894th component which is oneA soft demapper is then used to compute LLRs according
to (25) We refer to this algorithm as improved VBLAST (I-VBLAST) in the sequel
44 Low-Complexity K-Best Decoder The low-complexityK-Best (LC-K-Best) decoder recently proposed in [20] usestwo improvements over the classical K-Best decoder for thesake of lower complexity and latency The first improvementsimplifies the hybrid enumeration of the constellation pointsin real-valued systemmodel when the a priori information isincorporated into the tree search using two look-up tablesThe second improvement is to use a relaxed on-demandexpansion that reduces the need of exhaustive expansionand sorting operations The LC-K-Best algorithm can bedescribed as follows
The preprocessing step is as follows
(1) Input yH 119870CalculateH = QR y = Q119867y
(2) Enumerate the constellation symbols based on119898119860 for
all layers
The tree-search step is as follows
(1) Set layer to 2119873119905 listL = 0 119889
2119873119905+1
= 0
(a) expand all radic2119876 possible constellation nodes(b) calculate the corresponding PEDs(c) if radic2119876 gt 119870 select the 119870 best nodes and store
them in the listL2119873119905
(2) For layer 119894 = 2119873119905minus 1 1
(a) enumerate the constellation point according to119898119862
119894of the 119870 surviving paths in the listL
119894+1
(b) find the first child (FC) based on 119898119862
119894and 119898
119860
119894for
each parent nodes(c) compute their PEDs(d) select 119860 best children with smallest PEDs
among the 119870 FCs and add them to the listL119894
(e) if |L119894| lt 119870 find the next child (NC) of the
selected parent nodesCalculate their PEDs and go to step (2)(d)
(f) else move to the next layer 119894 = 119894 minus 1 and go tostep (2)
(3) If 119894 = 1 calculate the LLR as in (19)In the case of missing counter hypothesis LLR clip-ping method is used
It has been shown in [20] that the LC-K-Best decoderachieves almost the same performance as the classical K-Best decoder with different modulations Moreover thecomputational complexity in terms of the number of visitednodes is significantly reduced specially in the case of high-order modulations
5 Convergence of IterativeDetection-Decoding
The EXtrinsic Information Transfer (EXIT) chart is a usefultool to study the convergence behavior of iterative decod-ing systems [50] It describes the exchange of the mutualinformation in the iterative process in order to predict therequired number of iterations the convergence threshold(corresponding to the start of the waterfall region) and theaverage decoding trajectory
In the iterative receiver considered in our study two itera-tive processes are performed one inside the channel decoder(turbo or LDPC) and the other between the MIMO detectorand the channel decoder For simplicity we separately studythe convergence of the channel decoding and the MIMOdetection We denote by 119868
1198601and 119868
1198602the a priorimutual input
information of the MIMO detector and the channel decoderrespectively and by 119868
1198641and 119868
1198642their corresponding extrinsic
mutual output informationThe mutual information 119868
119909(119868119860or 119868
119864) can be computed
through Monte Carlo simulation using the probability den-sity function 119901
119871119909
[50]
119868119909
=1
2sum
119909isinminus11
int
+infin
minusinfin
119901119871119909
(119871119909
| 119909)
sdot log2
2119901119871119909
(119871119909
| 119909)
119901119871119909
(119871119909
| minus1) 119901119871119909
(119871119909
| +1)119889119871
119909
(29)
A simple approximation of the mutual information is used inour analysis [51]
119868119909
asymp 1 minus1
119871119887
119871119887minus1
sum
119899=0
log2
(1 + exp (minus119909119871119909)) (30)
where 119871119887is the number of transmitted bits and 119871
119909is the LLR
associated with the bit 119909 isin minus1 +1The a priori information 119871
119860can be modeled by applying
an independentGaussian randomvariable 119899119860with zeromean
and variance 1205902
119860in conjunction with the known transmitted
information bits 119909 [50]
119871119860
= 120583119860
119909 + 119899119860
where 120583119860
=1205902
119860
2 (31)
For each given mutual information value 119868119860
isin [0 1] 1205902
119860can
be computed using the following equation [52]
120590119860
asymp (minus1
1198671
log2
(1 minus 11986811198673
119860))
121198672
(32)
where 1198671
= 03073 1198672
= 08935 and 1198673
= 11064At the beginning the a priori mutual information is as
follows 1198681198601
= 0 and 1198681198602
= 0 Then the extrinsic mutualinformation 119868
1198641of the MIMO detector becomes the a priori
mutual information 1198681198602
of the channel decoder and so onand so forth (ie 119868
1198641= 119868
1198602and 119868
1198642= 119868
1198601) For a successful
decoding there must be an open tunnel between the curves
Mobile Information Systems 9
Table 1 Simulation parameters for convergence analysis
MIMO system 4 times 4 spatial multiplexingModulation 16-QAM Gray mappingChannel Flat Rayleigh fading
DetectorSTS-SD 119871 clip = plusmn8
LC-119870-Best 119870 = 16 119871 clip = plusmn3
MMSE-IC
Channel decoder
LTE turbo code [13 15]119900
Block length 119870119887
= 1024 119877119888
= 12
LDPC code (IEEE 80211n)Codeword length 119873
119887= 1944 119877
119888= 12
Interleaver Random size = 1024 (turbo code case)
the exchange of extrinsic information can be visualized as aldquozigzagrdquo decoding trajectory in the EXIT chart
To visualize the exchange of extrinsic information of theiterative receiver we present the MIMO detector and thechannel decoder characteristics into a single chart For ourconvergence analysis a 4 times 4 MIMO system with 16-QAMconstellation turbo decoder and LDPCdecoder (119877
119888= 12) is
considered Table 1 summarizes the main parameters for theconvergence analysis
Figure 5 shows the extrinsic information transfer char-acteristics of MIMO detectors at different 119864
119887119873
0values
As the I-VBLAST detector performs successive interferencecancellation at the first iteration and parallel interference can-cellation of the soft estimated symbols for the rest iterationsit is less intuitive to present its convergence in the EXITchart Therefore the convergence analysis of VBLAST is notconsidered
It is obvious that the characteristics of the detectorsare shifted upward with the increase of 119864
119887119873
0 We show
that for low 119864119887119873
0(0 dB) and for low mutual information
(lt01) MMSE-IC performs better than LC-K-Best decoderHowever with larger mutual information its performance islower Moreover the mutual information of STS-SD is higherthan LC-K-Best decoder and MMSE-IC for different 119864
119887119873
0
For higher 119864119887119873
0(5 dB) MMSE-IC presents lower mutual
information than other decoders when 1198681198601
lt 09Figure 6 shows the EXIT chart for 119864
119887119873
0= 2 dB with
several MIMO detectors namely STS-SD LC-K-Best andMMSE-IC We note that the characteristic of the channeldecoder is independent of 119864
119887119873
0values It is obvious that
the extrinsic mutual information of the channel decoderincreases with the number of iterations We see that after 6 to8 iterations in the case of turbo decoder (Figure 6(a)) thereis no significant improvement on the mutual informationMeanwhile in the case of LDPC decoder in Figure 6(b) 20iterations are enough for LDPC decoder to converge
By comparing the characteristics of STS-SD LC-K-Bestdecoder andMMSE-IC equalizer with both coding schemeswe notice that STS-SD has a larger mutual informationat its output LC-K-Best decoder has slightly less mutualinformation than STS-SD MMSE-IC shows low mutualinformation levels at its output compared to other algorithms
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
STS-SDLC-K-Best
MMSE-IC
EbN0 = 0 2 5dB
5dB
2dB
0dBI E1I
A2
IA1 IE2
Figure 5 Extrinsic information transfer characteristics of MIMOdetectors (STS-SD LC-K-Best andMMSE-IC) at119864
119887119873
0= 0 2 5 dB
in a 4 times 4 MIMO system using 16-QAM
when 1198681198601
lt 09 while for 1198681198601
gt 09 the extrinsic mutualinformation is comparable to others
In the case of turbo decoder (Figure 6(a)) with 119868in =
8 3 outer iterations are sufficient for STS-SD to convergeat 119864
119887119873
0= 2 dB However the same performance can
be attained by performing 4 outer iterations with only 2inner iterations Similarly LC-K-Best decoder shows anequivalent performance but slightly higher119864
119887119873
0is required
The convergence speed of LC-K-Best decoder is a bit lowerthan STS-SD which requires more iterations to get the sameperformance The reason is mainly due to the unreliabilityof LLRs caused by the small list size (L = 16) In thecase of MMSE-IC the characteristic presents a lower mutualinformation than the LC-K-Best decoder when 119868
1198601lt 09
Therefore an equivalent performance can be obtained athigher 119864
119887119873
0or by performing more iterations
In a similar way we study the convergence of MIMOdetection algorithms with LDPC decoder Figure 6(b) showsthe EXIT chart at 119864
119887119873
0= 2 dBThe same conclusion can be
retrieved as in the case of turbo decoder We can see that at119864119887119873
0= 2 dB a clear tunnel is observed between the MIMO
detector and the channel decoder characteristics allowingiterations to bring improvement to the system SimilarlySTS-SD offers higher mutual information than LC-K-Bestdecoder andMMSE-IC equalizer which suggests its superiorsymbol detection performance
Additionally the average decoding trajectory resultingfrom the free-run iterative detection-decoding simulationsis illustrated in Figure 6 at 119864
119887119873
0= 2 dB with 119868out = 4
119868in = 2 and 119868out = 4 119868in = 5 in the case of turbo decoder andLDPC decoder respectively The decoding trajectory closely
10 Mobile Information Systems
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
TurboSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 2 6 8
(a)
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
LDPCSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 5 10 20
(b)
Figure 6 EXIT chart for MIMO detectors (STS-SD LC-K-Best and MMSE-IC) and channel decoder (a) LTE turbo decoder 119870119887
= 1024119877119888
= 12 and (b) LDPC decoder 119873119887
= 1944 119877119888
= 12 at 119864119887119873
0= 2 dB in a 4 times 4 MIMO system using 16-QAM
matches the characteristics in the case of STS-SD and LC-K-Best decoders The little difference from the characteristicsafter a few iterations is due to the correlation of extrinsicinformation caused by the limited interleaver depth In thecase of MMSE-IC the decoding trajectory diverges fromthe characteristics for high mutual information because theequalizer uses the a posteriori information to compute softsymbols instead of the extrinsic information
The best trade-off scheduling of the required numberof iterations is therefore 119868out iterations in the outer loopand a total of 8 iterations inside the turbo decoder and 20iterations inside the LDPC decoder distributed across these119868out iterations
6 Performance and Complexity Evaluation ofIterative Detection-Decoding
In this section we evaluate and compare the performance andthe complexity of differentMIMOdetectors namely STS-SDLC-K-Best decoder MMSE-IC and I-VBLAST equalizerswith different channel coding techniques (turbo LDPC)A detailed analysis of the performance and the complexitytrade-off ofMIMOdetection with LTE turbo decoder and 16-QAM modulation in a Rayleigh channel has been discussedin [24] Herein the performance and the complexity ofthe receiver with LDPC decoder are investigated Moreoverseveral modulations and coding schemes are consideredto quantify the gain achieved by such iterative receiver indifferent channel environments Consequently a comparative
study is conducted in iterative receiver with both codingschemes (turbo LDPC)
For the turbo code the rate 13 turbo encoder specifiedin 3GPP LTE with a block length 119870
119887= 1024 is used in the
simulations Puncturing is performed in the rate matchingmodule to achieve an arbitrary coding rate 119877
119888(eg 119877
119888= 12
34)Meanwhile the LDPC encoder specified in IEEE 80211nis consideredThe encoder is defined by a parity checkmatrixthat is formed out of square submatrices of sizes 27 54 or 81Herein the codeword length of size 119873
119887= 1944 with coding
rate of 119877119888
= 12 and 34 is considered
61 Performance Evaluation The simulations are first carriedout in Rayleigh fading channel to view general performanceof the iterative receivers Real channel models will be consid-ered to evaluate the performance in more realistic scenariosTherefore the 3GPP LTE-(A) channel environments withlow medium and large delay spread values and Dopplerfrequencies are considered The low spread channel is theExtended Pedestrian A (EPA) model which emulates theurban environment with small cell sizes (120591rms = 43 ns) Themedium spread channel (120591rms = 357 ns) is the ExtendedVehicular A (EVA) model The Extended Typical Urban(ETU) model is the large spread channel which has alarger excess delay (120591rms = 991 ns) and simulates extremeurban suburban and rural cases Table 2 summarizes thecharacteristic parameters of these channel environments Forall cases the channel is assumed to be perfectly known atthe receiver Table 3 gives the principle parameters of thesimulations The performance is measured in terms of bit
Mobile Information Systems 11
Table 2 Characteristic parameters of the investigated channelmodels
error rate (BER) with respect to signal-to-noise ratio (SNR)per bit 119864
119887119873
0
119864119887
1198730
=119864119904
1198730
+ 10 sdot log10
1
119877119888119876119873
119905
[dB] (33)
In our previous study [24] the performance of MIMOdetectors with LTE turbo decoder is evaluated in a Rayleighchannel with various outer and inner iterations It has beenshown that the performance is improved by about 15 dB at aBER level of 1 times 10
minus5 with 4 outer iterations It has been alsoshown that no significant improvement can be achieved after4 outer iterations this improvement is less than 02 dB with 8outer iterations
Similarly to turbo decoder we fix the number of inneriterations inside LDPC decoder to 20 while varying thenumber of outer iterations Figure 7 shows BER performanceof MIMO detectors with LDPC decoder in Rayleigh channelwith 119868in = 20 iterations and 119868out = 1 2 4 or 8 iterationsSTS-SD is used with a LLR clipping level of 8 which givesclose toMAPperformancewith considerable reduction in thecomplexity For LC-K-Best decoder a LLR clipping level of 3is used in the case of missing counter hypothesis We show
0 1 2 3 4 5
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
Iout = 4
Iout = 8
Iout = 2
Iout = 1
EbN0 (dB)minus1
Figure 7 BER performance of a 4 times 4 spatial multiplexing systemwith 16-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) in a Rayleigh fading channel 119868out = 1 2 4 8 119868in =
20 LDPC decoder 119873119887
= 1944 and 119877119888
= 12
that performance improvement of 15 dB is observed with 4outer iterations For 119868out = 8 iterations the improvementis less than 02 dB Therefore 119868out = 4 iterations will beconsidered in the sequel
Figure 8 shows the BER performance of 4 times 4 16-QAMsystem in a Rayleigh fading channel with 119868out = 4 and 119868in = 2
in the case of turbo decoder and 119868in = [3 4 6 7] in the caseof LDPC decoder The notation 119868in = [3 4 6 7] denotes that3 inner iterations are performed in the 1st outer iteration 4inner iterations in the 2nd outer iteration and so on Theperformances of STS-SD with 119868in = 8 and 119868in = 20 foreach outer iteration in the case of turbo decoder and LDPCdecoder are also plotted as a reference In the case of turbodecoder we show that performing 119868in = 8 and 119868out = 4
iterations does not bring significant improvement on theperformance compared to the case when 119868in = 2 and 119868out =
4 iterations are performed Similarly in the case of LDPCdecoder the performance of 119868in = 20 and 119868out = 4 iterationsis comparable to the performance of 119868in = [3 4 6 7] and119868out = 4 iterations Hence using a large number of iterationsdoes not seem to be efficientwhich proves the results obtainedin the convergence analysis of Section 5
By comparing the algorithms LC-K-Best decoder showsa degradation of less than 02 dB compared to STS-SD ata BER level of 2 times 10
minus5 However it outperforms MMSE-IC and I-VBLAST equalizer by about 02 dB at a BERlevel of 2 times 10
minus5 MMSE-IC and I-VBLAST show almostthe same performance In addition in the case of LDPCdecoder (Figure 8(b)) we notice that increasing the numberof inner iterations for each outer iteration 119868in = [3 4 6 7]
12 Mobile Information Systems
0 05 1 15 2 25 3
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 8
(a) LTE turbo decoder
BER
0 05 1 15 2 25 3
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 20
STS-SD Iin = 5
STS-SD
LC-K-BestI-VBLASTMMSE-IC
(b) LDPC decoder
Figure 8 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
shows slightly better performance than performing an equalnumber of iterations 119868in = 5 for each outer iteration
In the case of high-order modulation higher spectralefficiency can be achieved at a cost of increased symboldetection difficulty Figure 9 shows the BER performance of64-QAM with 119877
119888= 34 We see that LC-K-Best decoder
with a list size of 32 presents the similar performance as STS-SD However I-VLAST equalizer and MMSE-IC equalizerpresent degradation of more than 2 dB at a BER level of 1 times
10minus5 compared to LC-K-Best decoder Therefore LC-K-Best
decoder is more robust in the case of high-ordermodulationsand high coding rates The figure also shows that the BERperformance of LDPC decoder is almost identical to that ofthe turbo decoder
In order to summarize the performance of differentdetectors with different channel decoders we provide the119864119887119873
0values achieving a BER level of 1 times 10
minus5 in Table 4The values given in the parentheses of the table represent theperformance loss compared to STS-SD
Next we evaluate the performance of the iterative receiverin more realistic channel environments Figures 10 11 and 12show the BER performance of the detectors with the channeldecoders on EPA EVA and ETU channels receptivelySimilar behaviors can be observed with LTE turbo decoderand with LDPC decoder
In EPA channel (Figure 10) we see that LC-K-Bestdecoder achieves similar performance as STS-SD in the caseof 64-AQM and presents a degradation less than 02 dB in the
case of 16-QAM Meanwhile MMSE-IC presents significantperformance loss of more than 6 dB in the case of 64-QAMand 119877
119888= 34 With 16-QAM and 119877
119888= 12 the degradation
of MMSE-IC compared to LC-K-Best decoder is about 1 dBat a BER level of 1 times 10
minus4In EVA channel (Figure 11) the performance loss of
MMSE-IC compared to LC-K-Best decoder is reduced toapproximately 5 dB with 64-QAM and 05 dB with 16-QAMLC-K-Best decoder presents a degradation of about 01sim03 dB compared to STS-SD in the case of 16-QAM and 64-QAM
Similarly in ETU channel (Figure 12) MMSE-IC presentsa performance degradation compared to LC-K-Best decoderThis degradation is less than 4 dB in the case of 64-QAM andless than 05 dB in the case of 16-QAM We notice also thatthe LC-K-Best decoder is comparable to STS-SD in the caseof 64-QAM and has a degradation of 02 dB in the case of 16-QAM
Comparing the performance of the iterative receiver indifferent channels it can be seen that iterative receiverspresent the best performance in ETU channel compared toEPA and EVA channels This is due to the high diversity ofETU channel At a BER level of 1 times 10
minus4 the performancegain in ETU channel in the case of LTE turbo decoder is about08 dB 13 dB compared to EVA channel with 16-QAM and64-QAM respectively In the case of LDPC decoder this gainis 04 dB and 1 dB with 16-QAM and 64-QAM respectivelyHowever in EPA channel the performance gain in ETU
Mobile Information Systems 13
3 4 5 6 7 8 9 10 11 12 13
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
EbN0 (dB)
(a) LTE turbo decoder
3 4 5 6 7 8 9 10 11 12
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)
STS-SD Iin = 20
STS-SDLC-K-Best
I-VBLASTMMSE-IC
(b) LDPC decoder
Figure 9 BER performance of a 4 times 4 spatial multiplexing system with 64-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
0values achieving a BER level of 1times 10minus5 for different detectors and channel decoders (turbo LDPC) in 4times 4 spatialmultiplexing
system with 16-QAM 119877119888
= 12 and 64-QAM 119877119888
= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
STS-SD 25 dB 102 dB 27 dB 96 dBLC-119870-Best 26 dB (minus01) 102 dB (00) 28 dB (minus01) 96 dB (00)I-VBLAST 28 dB (minus03) 128 dB (minus26) 29 dB (minus02) 12 dB (minus24)MMSE-IC 28 dB (minus03) 128 dB (minus26) 30 dB (minus03) 12 dB (minus24)lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
channel in the case of turbo decoder or LDPC decoder ismore than 1 dB with 16-QAM and 64-QAM
Table 5 summarizes the 119864119887119873
0values achieving a BER
level of 1times10minus4 of different detectors combined with different
channel decoders andmodulation orders in different channelmodels The values given in the parentheses in the tablerepresent the performance loss compared to STS-SD Asindicated in Table 5 the iterative receivers with turbo decoderand LDPC decoder have comparable performance with acoding rate 119877
119888= 12 (16-QAM) However with 119877
119888= 34
(64-QAM) the receivers with LDPC decoder present slightlya better performance especially in ETU channel (06 dB)
From these results we show that the iterative receiversubstantially improves the performance of coded MIMOsystems either with turbo decoder or with LDPC decoderin Rayleigh channel (Figures 8 and 9) and in more realisticchannels (Figures 10 11 and 12) Moreover we show that
performing a large number of inner iterations does notbring significant improvement In addition we show thatthe LC-K-Best decoder achieves a good performance withdifferent modulations and channel coding schemes Thefigures suggest that the BER performance of the iterativereceiver with turbo decoder is almost comparable to that ofthe LDPC decoder It is therefore meaningful to evaluate thecomputational complexity of the iterative receivers with bothdecoding techniques as it will be discussed in the next section
62 Complexity Evaluation The computational complexityhas significant impact on the latency throughput and powerconsumption of the deviceTherefore the receiver algorithmsshould be optimized to achieve a good trade-off betweenperformance and cost In this part we evaluate the com-putational complexity of the iterative receiver in terms ofbasic operations such as addition subtractionmultiplication
14 Mobile Information Systems
1 3 5 7 9 11 13 15 17 19
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
(a) EPA turbo 119868out = 4 119868in = 2
BER
1 3 5 7 9 11 13 15 17 19
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12 64-QAM
Rc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EPA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 10 BER performance of a 4 times 4 spatial multiplexing with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) on EPA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944 119868out = 4
119868in = [3 4 6 7]
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
STS-SDLC-K-Best
MMSE-IC
16-QAMRc = 12
64-QAMRc = 34
(a) EVA turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EVA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 11 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Best and MMSE-IC) on EVA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944
and 119868out = 4 119868in = [3 4 6 7]
Mobile Information Systems 15
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(a) ETU turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) ETU LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 12 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SDLC-K-Best and MMSE-IC) on ETU channel model (a) LTE turbo decoder 119870
minus4 in LTE channel models for different detectors and channel decoders (turbo LDPC)in 4 times 4 spatial multiplexing system with 16-QAM 119877
119888= 12 and 64-QAM 119877
119888= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
EPASTS-SD 63 dB 140 dB 62 dB 138 dB
LC-119870-Best 65 dB (minus02) 139 dB (+01) 64 dB (minus02) 139 dB (minus01)MMSE-IC 74 dB (minus11) gt20 dB (gtminus6) 80 dB (minus18) gt20 dB (gtminus6)
EVASTS-SD 52 dB 143 dB 53 dB 134 dB
LC-119870-Best 54 dB (minus02) 144 dB (minus01) 56 dB (minus03) 137 dB (minus03)MMSE-IC 58 dB (minus06) 190 dB (minus47) 60 dB (minus07) 185 dB (minus51)
ETUSTS-SD 44 dB 130 dB 49 dB 124 dB
LC-119870-Best 46 dB (minus02) 130 dB (00) 49 dB (00) 124 dB (00)MMSE-IC 49 dB (minus05) 175 dB (minus35) 51 dB (minus02) 16 dB (minus36)
lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
division square root extraction maximization and look-up table check (which are denoted by ADD SUB MULDIV SQRT Max and LUT resp) To this end the completecomparison of the iterative receivers with both channeldecoders (turbo LDPC) and with several modulations andcoding rates is carried out
621 Complexity of Iterative Receiver The complexity ofiterative receiver depends on theMIMOdetector the channeldecoder and the number of innerouter iterationsThis com-plexity can be expressed by
sdot [119862det1 + (119868out minus 1) sdot 119862deti] (34)
where 119862det1 denotes the complexity of the first iterationof MIMO detection algorithm per symbol vector withouttaking into consideration the a priori information 119862detidenotes the complexity per iteration per symbol vector takinginto consideration the a priori information 119862dec denotesthe complexity of the channel decoder per iteration perinformation bit 119873bit is the number of information bit at theinput of the encoder 119873symb is the number of symbol vectors119873symb and 119873bit are linked by the following relation
119873symb =119873bit
119876119877119888119873119905
= 120572119873bit with 120572 =1
119876119877119888119873119905
(35)
where 119876 is the number of bits in the constellation symbol 119877119888
is the coding rate and119873119905is the number of transmit antennas
16 Mobile Information Systems
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
Figure 1 Block diagram of MIMO-OFDM system using bit-interleaved coded modulation with iterative detection and decoding
22 Iterative Detection-Decoding Principle At the receiver torecover the transmitted signal from interferences an iterativedetection-decoding process based on the turbo principle isapplied as depicted in Figure 1 The MIMO detector andthe channel decoder exchange soft information that is loglikelihood ratio (LLR) in each iteration
The MIMO detector takes the received symbol vector yand the a priori information 119871
1198601of the coded bits from the
channel decoder and computes the extrinsic information1198711198641
The MIMO detection algorithm can be the MAP algorithmor other suboptimal algorithms like STS-SDK-Best decoderI-VBLAST or MMSE-IC The extrinsic information is dein-terleaved and becomes the a priori information 119871
1198602for the
channel decoderThe channel decoder computes the extrinsicinformation 119871
1198642that is reinterleaved and fed back to the
detector as the a priori information 1198711198601
The channel decoding is performed either by an LTEturbo decoder or by an LDPC decoder which exchanges softinformation between their component decoders as describedin the next section In our iterative process we denote thenumber of outer iterations between the MIMO detector andthe channel decoder by 119868out and the number of iterationswithin the turbo decoder or LDPC decoder by 119868in
For QAM the mapping process can be done inde-pendently for real and imaginary part The system modelexpressed in (2) can be converted into an equivalent real-valued model
[
Re (y)
Im (y)] = [
Re (H) minus Im (H)
Im (H) Re (H)] [
Re (s)Im (s)
]
+ [
Re (n)
Im (n)]
(3)
where Re(sdot) and Im(sdot) represent the real and imaginary partsof a complex number respectively Each QAM constellationpoint is treated as two PAM symbols and the matrix dimen-sion is doubled However as shown in [17] the real-valuedmodel is more efficient for the implementation of the sphere
decoder Hence it will be used as the systemmodel in case ofsphere decoding in the following sections
3 Soft-Input Soft-Output Channel Decoder
Channel coding is used to protect the useful informationfrom channel distortion and noise by introducing someredundancy The state-of-the-art channel coding schemessuch as the LDPC [26] and turbo codes [3] can effectivelyapproach the Shannon bound LDPC codes are nowadaysadopted in many standards including IEEE 80211 and DVB-T2 as they achieve very high throughput due to inherentparallelism of the decoding algorithm In the meantime theturbo codes are also adopted in LTE LTE-A (binary turbocodes) and WiMAX (double binary turbo codes) In thispaper LDPC codes and LTE turbo codes are considered
31 Turbo Decoder Initially proposed in 1993 [3] turbocodes have attracted great attention due to the capacity-approaching performance The turbo encoder is constitutedby a parallel concatenation of two recursive systematicconvolutional encoders separated by an interleaver Thefirst encoder processes the original data while the secondprocesses the interleaved version of dataThemain role of theinterleaver is to reduce the degree of correlation between theoutputs of the component encoders
In LTE system the recursive systematic encoders with8 states and [13 15]
119900polynomial generators are adopted A
quadratic polynomial permutation (QPP) interleaver is usedas a contention free interleaver and it is suitable for paralleldecoding of turbo codes as illustrated in Figure 2(a) Themother coding rate is 13 Coding rates other than themotherrate can be achieved by puncturing or repetition using the ratematching technique
The turbo decoding is performed by two SISO compo-nent decoders that exchange soft information of their datasubstreams Each component decoder takes systematic orinterleaved information the corresponding parity informa-tion and the a priori information from the other decoder to
4 Mobile Information Systems
LTE QPPinterleaver
D D D
DDD
uk (Info)
1st encoder
2nd encoder
sk
p1k
p2k
s998400k
⨁
⨁ ⨁
⨁
⨁
⨁
⨁
⨁
(a) Turbo encoder
DEC2
DEC1
Lc(in) SP
Lc(p1)
Lc(s)
Lc(p2)
Πi
La(u)
Πminus1i ΠiIin
Le(u)
Le(u)
Le(u)
La(u)
Le(p2)
Le(p1)
PSLc(out)
(b) Turbo decoder
Figure 2 Structure of LTE turbo code with 119877119888
= 13 (a) turboencoder and (b) turbo decoder
compute the extrinsic information as shown in Figure 2(b)Two families of decoding algorithms can be used soft-output Viterbi algorithms (SOVA) [27 28] and maximum aposteriori (MAP) algorithm [29] The MAP algorithm offerssuperior performance but suffers from high computationalcomplexity Two suboptimal algorithms namely log-MAPand max-log-MAP are practically used [30] Herein log-MAPalgorithm is considered by using the Jacobian logarithm[30]
where 119891119888(|119909 minus 119910|) is a correction function that can be
computed using a small look-up table (LUT)The decoder computes the branch metrics (120574) and the
forward (120572) and the backward (120573) metrics between two statesin the trellis as follows
120574119896
(119904119896minus1
119904119896) = 119901 (119904
119896 119910119896
| 119904119896minus1
)
120572119896
(119904119896) =
lowastmax119904119896minus1
(120572119896minus1
(119904119896minus1
) + 120574119896
(119904119896minus1
119904119896))
120573119896
(119904119896) =
lowastmax119904119896+1
(120573119896+1
(119904119896+1
) + 120574119896
(119904119896 119904119896+1
))
(5)
The a posteriori LLRs of the information bits are computed as
119871 (119906119896) =
lowastmax119906119896=0
(120572119896minus1
(119904119896minus1
) + 120574119896
(119904119896minus1
119904119896) + 120573
119896(119904119896))
minuslowastmax119906119896=1
(120572119896minus1
(119904119896minus1
) + 120574119896
(119904119896minus1
119904119896) + 120573
119896(119904119896))
(6)
The component decoders exchange only the extrinsic LLRwhich is defined by
119871119890(119906119896) = 119871 (119906
119896) minus 119871
119886(119906119896) minus 119871
119904(119906119896) (7)
where 119871119886(119906119896) and 119871
119904(119906119896) correspond to the a priori informa-
tion from the other decoder and the systematic informationbits respectively
32 LDPC Decoder LDPC codes belong to a class of linearerror correcting block codes first proposed by Gallager [26]Their main advantages lie in their capacity-approaching per-formance and their low-complexity parallel implementations[31] LDPC codes can be represented by a parity checkmatrix HLDPC or intuitively through Tanner graph [32]Tanner graphs are bipartite graphs containing two types ofnodes the check nodes and the variable nodes as illustratedin Figure 3(a) It consists of 119872 check nodes (CN) whichcorrespond to the number of parity bits (ie number of rowsof HLDPC) and 119873 variable nodes (VN) corresponding to thenumber of bits in a codeword (ie number of columns ofHLDPC)
The optimal maximum a posteriori decoding of LDPCcodes is infeasible from the practical implementation point ofviewAlternatively LDPCdecoding is done using themessagepassing or belief propagation algorithms which iterativelypass messages between check nodes and variable nodes asshown in Figure 3(b) The belief propagation is denotedas the sum-product decoding because probabilities can berepresented as LLRs which allow the calculation of messagesusing sum and product operations
Let 119871V119894119895
be themessage from variable node 119894 to check node119895 and 119871
119888119895119894
the message from check node 119895 to variable node119894 Let 119881
119895and 119862
119894denote the set of adjacent variable nodes
connected to the check node 119895 and the set of adjacent checknodes connected to the variable node 119894 respectively where0 le 119895 le 119872 and 0 le 119894 le 119873 For the first iteration theinput to the LDPC decoder is the LLRs 119871(119888
119894) of the codeword
c which are used as an initial value of the extrinsic variablenode messages that is 119871V
119894119895
= 119871(119888119894) For the 119896th iteration the
algorithm can be summarized as follows(1) Each check node 119895 computes the extrinsic message to
its neighboring variable node 119894
119871119888119895119894
= 2tanhminus1 ( prod
1198941015840isin119881119895119894
tanh(
119871V1198941015840119895
2)) (8)
(2) Each variable node 119894 updates its extrinsic informationto the check node 119895 in the next iteration
119871V119894119895
= 119871 (119888119894) + sum
1198951015840isin119862119894119895
1198711198881198951015840119894
(9)
Mobile Information Systems 5
n = 6m = 4
VN (n)
CN (m)
0 1 1 0 0 1
1 1 1 0 1 0
1 0 0 1 1 1
0 0 1 1 0 1
=HLDPC
(a)
1 2 3 5 64
1 2 3 4
VNi
CNj
L13
L12
L31
L32
L34
Lc12Lc13
Lc16Lc31
Lc34
Lc35
Lc36
(b)
Figure 3 LDPC decoder (a) matrix representation and Tanner graph and (b) iterative decoding between VNs and CNs
(3) The a posterioriLLRof each codeword bit is computedas
119871119901
(119888119894) = 119871 (119888
119894) + sum
1198951015840isin119862119894
1198711198881198951015840119894
(10)
The decoding algorithm alternates between check node proc-essing and variable node processing until a maximum num-ber of iterations are achieved or until the parity check con-dition is satisfied When the decoding process is terminatedthe decoder outputs the a posteriori LLR
4 Soft-Input Soft-Output MIMO Detection
The aim of MIMO detection is to recover the transmittedvector s from the received vector y The state-of-the-artMIMO detection algorithms have been presented in [24]These algorithms can be divided into two main familiesnamely the tree-search-based detection and the interference-cancellation-based detection In this section we brieflyreview the main existing SISO MIMO detection algorithmsuseful for the following sections
41 MaximumA Posteriori Probability (MAP) Detection TheMAP algorithm achieves the optimum performance throughthe use of an exhaustive search over all 2
119876sdot119873119905 possible symbol
combinations to compute the LLR of each bit The LLR of the119887th bit in the 119894th transmit symbol 119909
119894119887 is given by
119871 (119909119894119887
) = log119875 (119909
119894119887= +1 | y)
119875 (119909119894119887
= minus1 | y)
= logsumsisin120594+1
119894119887
119901 (y | s) 119875 (s)
sumsisin120594minus1119894119887
119901 (y | s) 119875 (s)
(11)
where 120594+1
119894119887and 120594
minus1
119894119887denote the sets of symbol vectors in
which the 119887th bit in the 119894th antenna is equal to +1 and minus1respectively 119901(y | s) is the conditioned probability densityfunction given by
119901 (y | s) =1
(1205871198730)119873119903
exp(minus1
1198730
1003817100381710038171003817y minus Hs10038171003817100381710038172
) (12)
119875(s) represents the a priori information provided by thechannel decoder in the form of a priori LLRs
119871119860
(119909119894119887
) = log119875 (119909
119894119887= +1)
119875 (119909119894119887
= minus1) forall119894 119887
119875 (s) =
119873119905
prod
119894=1
119875 (119904119894) =
119873119905
prod
119894=1
119876
prod
119887=1
119875 (119909119894119887
)
(13)
The max-log-MAP approximation is commonly used in theLLR calculation with lower complexity [12]
119871 (119909119894119887
) asymp1
1198730
min120594minus1
119894119887
1198891 minus
1
1198730
min120594+1
119894119887
1198891 (14)
1198891
=1003817100381710038171003817y minus Hs1003817100381710038171003817
2
minus 1198730
119873119905
sum
119894=1
119876
sum
119887=1
log119875 (119909119894119887
) (15)
where 1198891represents the Euclidean distance between the
received vector y and lattice pointsHsBased on the a posteriori LLRs 119871(119909
119894119887) and the a priori
LLRs 119871119860
(119909119894119887
) the detector computes the extrinsic LLRs119871119864(119909119894119887
) as
119871119864
(119909119894119887
) = 119871 (119909119894119887
) minus 119871119860
(119909119894119887
) (16)
The MAP algorithm is not feasible due to its exponentialcomplexity since 2
119876sdot119873119905 hypotheses have to be considered
within each minimum term and for each bit Thereforeseveral suboptimal MIMO detectors have been proposedwith reduced complexity as will be briefly discussed in thefollowing sections
42 Tree-Search-Based Detection The tree-search-baseddetection methods generally fall into two main categoriesnamely depth-first search like the sphere decoder andbreadth-first search like the K-Best decoder
421 List Sphere Decoder (LSD) The basic idea of the spheredecoder is to limit the search space of the MAP solution to ahypersphere of radius 119903
119904around the received vector Instead
of testing all the hypotheses of the transmitted signal onlythe lattice points that lie inside the hypersphere are testedreducing the computational complexity [33]
sSD = arg minsisin2119876119873119905
1003817100381710038171003817y minus Hs1003817100381710038171003817
2
le 1199032
119904 (17)
6 Mobile Information Systems
Using the QR decomposition in real-valued model thechannel matrix H can be decomposed into two matrixes Qand R (H = QR) where Q is 2119873
119903times 2119873
119905orthogonal matrix
(Q119867Q = I2119873119905
) and R is 2119873119905
times 2119873119905upper triangular matrix
with real-positive diagonal elements [34] Therefore the dis-tance in (17) can be computed as yminusHs2 = yminusRs2 wherey = Q119867y is the modified received symbol vector Exploitingthe triangular nature of R the Euclidean distance metric 119889
1
in (15) can be recursively evaluated through the accumulatedpartial Euclidean distance (PED) 119889
and the a priori-based partial metric at the 119894th level respec-tively
This process can be illustrated by a tree with 2119873119905
+ 1
levels as depicted in Figure 4(a) The tree search starts at theroot level with the first child node at level 2119873
119905 The partial
Euclidean distance 1198892119873119905
in (18) is then computed If 1198892119873119905
issmaller than the sphere radius 119903
119904 the search continues at level
2119873119905minus1 and steps down the tree until finding a valid leaf node
at level 1List sphere decoder is proposed to approximate the MAP
detector [12] It generates a list L sub 2119876119873119905 that includes the
best possible hypotheses The LLR values are then computedfrom this list as
119871 (119909119894119887
) =1
1198730
minLcap120594minus1119894119887
1198891 minus
1
1198730
minLcap120594+1119894119887
1198891 (19)
The main issue of LSD is the missing counter-hypothesisproblem depending on the list sizeThe use of limited list sizecauses inaccurate approximation of the LLR due to missingsome counter hypotheses where no entry can be found inthe list for a particular bit 119909
119894119887= +1 minus1 Several solutions
have been proposed to handle this issue LLR clipping is afrequently used solution which consists simply to set the LLRto a predefined maximum value [12 35]
Several methods can be considered to reduce the com-plexity of the sphere decoder such as Schnorr-Euchner(SE) enumeration [36] layer ordering technique [34] andchannel regularization [37] Layer ordering technique allowsthe selection of the most reliable symbols at a high layerusing the sorted QR (SQR) decomposition However channelregularization introduces a biasing factor in the metricswhich should be removed in LLR computation to avoidperformance degradation as discussed in [38] In the sequelthe SQR decomposition is considered in the preprocessingstep
422 Single Tree-Search SphereDecoder (STS-SD) Oneof thetwominima in (14) corresponds to theMAPhypothesis sMAPwhile the other corresponds to the counter hypothesis Thecomputation of LLR can be expressed by
119871 (119909119894119887
) =
1
1198730
(119889MAP119894119887
minus 119889MAP
) if 119909MAP119894119887
= +1
1
1198730
(119889MAP
minus 119889MAP119894119887
) if 119909MAP119894119887
= minus1
(20)
with
119889MAP
=10038171003817100381710038171003817y minus RsMAP10038171003817100381710038171003817
2
minus 1198730119875 (sMAP
)
119889MAP119894119887
= min119904isin120594
MAP119894119887
1003817100381710038171003817y minus Rs1003817100381710038171003817
2
minus 1198730119875 (s)
sMAP= arg min
119904isin2119876sdot119872
1003817100381710038171003817y minus Rs1003817100381710038171003817
2
minus 1198730119875 (s)
(21)
where 120594MAP119894119887
denotes the bitwise counter hypothesis of theMAP hypothesis which is obtained by searching over all thesolutions with the 119887th bit of the 119894th symbol opposite to thecurrentMAPhypothesisOriginally theMAPhypothesis andthe counter hypotheses can be found through repeating thetree search [39] The repeated tree search yields a large com-putational complexity cost To overcome this the single tree-search algorithm [13 40] was developed to compute all theLLRs concurrently The 119889
MAP metric and the corresponding119889MAP119894119887
metrics are updated through one tree-search processThrough the use of extrinsic LLR clipping method the STS-SD algorithm can be tunable between the MAP performanceand hard-output performance The implementations of STS-SD have been reported in [14 15]
423 SISO K-Best Decoder K-Best algorithm is a breadth-first search based algorithm in which the tree is traversedonly in the forward direction [41] This approach searchesonly a fixed number 119870 of paths with best metrics at eachdetection layer Figure 4(b) shows an example of the treesearch with 119873
119905= 2 The algorithm starts by extending the
root node to all possible candidates It then sorts the newpaths according to their metrics and retains the 119870 paths withsmallest metrics for the next detection layer
K-Best algorithm is able to achieve near-optimal perfor-mance with a fixed and affordable complexity for parallelimplementation Yet the major drawbacks ofK-Best decoderare the expansion and the sorting operations that are verytime consuming Several proposals have been drawn in theliterature to approximate the sorting operations such asrelaxed sorting [42] and distributed sorting [43] or evento avoid sorting using on-demand expansion scheme [44]Moreover similarly as LSD K-Best decoder suffers frommissing counter-hypothesis problem due to the limited listsize Numerous approaches have been proposed to addressthis problem such as smart candidates adding [45] bit flip-ping [46] and path augmentation and LLR clipping [12 35]
43 Interference-Cancellation-Based Detection Interference-cancellation-based detection can be carried out either in aparallel way as inMMSE-IC [8 9] or in a successive way as inVBLAST [47]
431 Minimum Mean Square Error-Interference Cancella-tion (MMSE-IC) Equalizer MMSE-IC equalizer can be per-formed using two filters [4]The first filter p
119894is applied to the
received vector y and the second filter q119894is applied to the
estimated vector s in order to cancel the interference fromother layers The equalized symbol
119894can be written as
119894= p119867
119894y minus q119867
119894s119894
with 119894 isin [1 119873119905] (22)
where s119894denotes the estimated vector given by the
previous iteration with the 119894th symbol omitteds119894
= [1 sdot sdot sdot 119894minus1
0 119894+1
sdot sdot sdot 119873119905]
119894is calculated by the
soft mapper as 119894
= E[119904119894] = sum
119904isin2119876 119904119875 (119904
119894= 119904) [48] The filters
p119894and q
119894are optimized using the MMSE criterion and are
given in [6 24]For the first iteration since no a priori information is
available the equalization process is reduced to the classicalMMSE solution
119894= (H119867H +
1205902
119899
1205902119904
I119873119905
)
minus1
H119867y (23)
The equalized symbols 119894are associated with a bias factor 120573
119894
in addition to some residual noise plus interferences 120578119894
119894= 120573
119894119904119894+ 120578
119894 (24)
These equalized symbols are then used by the soft demapperto compute the LLR values using the max-log-MAP approxi-mation [48]
119871 (119909119894119887
) =1
1205902120578119894
(min119904119894isin120594minus1
119894119887
1003816100381610038161003816119894 minus 120573119894119904119894
1003816100381610038161003816
2
minus min119904119894isin120594+1
119894119887
1003816100381610038161003816119894 minus 120573119894119904119894
1003816100381610038161003816
2
) (25)
MMSE-IC equalizer requires 119873119905matrix inversions for each
symbol vector For this reason several approximations of
MMSE-IC were proposed For example in [9] a low-complexity approach of MMSE-IC is described by perform-ing a single matrix inversion without performance loss Thisalgorithm is referred to as LC-MMSE-IC
432 Successive Interference Cancellation (SIC) EqualizerThe SIC-based detector was initially used in the VBLASTsystems In VBLAST architecture [47] a successive cancel-lation step followed by an interference nulling step is usedto detect the transmitted symbols However this methodsuffers from error propagation An improved VBLAST foriterative detection and decoding is described in [49] At thefirst iteration an enhanced VBLAST which takes decisionerrors into account is employed [24] When the a prioriLLRs are available from the channel decoder soft symbols arecomputed by a soft mapper and are used in the interferencecancellation To describe the enhanced VBLAST algorithmwe assume that the detection order has been made accordingto the optimal detection order [47] For the 119894th step thepredetected symbol vector s
119894minus1until step 119894 minus 1 is canceled out
from the received signal
y119894= y minus H
1119894minus1s119894minus1
(26)
where s119894minus1
= [1
2
sdot sdot sdot 119894minus1
] and H1119894minus1
=
[h1h2
sdot sdot sdot h119894minus1
] with h119894being the 119894th column of H
Then the estimated symbol 119894is obtained using a filtered
matrix W119894based on the MMSE criterion that takes decision
errors into account [11 49]
119894= W119867
119894y119894= 120573
119894119904119894+ 120578
119894
W119894= 120590
2
119904(HΣ
119894H119867 + 119873
0I119873119903
)minus1
h119894
(27)
Σ119894is the decision error covariance matrix defined as
Σ119894=
119894minus1
sum
119895=1
1205982
119895e119895e119879119895
+
119873119905minus119894+1
sum
119895=119894
1205902
119904e119895e119879119895
1205982
119895= E
10038161003816100381610038161003816119904119895
minus 119895
10038161003816100381610038161003816
2
| 119895minus1
(28)
8 Mobile Information Systems
where e119894denotes a unit vector having zero components except
the 119894th component which is oneA soft demapper is then used to compute LLRs according
to (25) We refer to this algorithm as improved VBLAST (I-VBLAST) in the sequel
44 Low-Complexity K-Best Decoder The low-complexityK-Best (LC-K-Best) decoder recently proposed in [20] usestwo improvements over the classical K-Best decoder for thesake of lower complexity and latency The first improvementsimplifies the hybrid enumeration of the constellation pointsin real-valued systemmodel when the a priori information isincorporated into the tree search using two look-up tablesThe second improvement is to use a relaxed on-demandexpansion that reduces the need of exhaustive expansionand sorting operations The LC-K-Best algorithm can bedescribed as follows
The preprocessing step is as follows
(1) Input yH 119870CalculateH = QR y = Q119867y
(2) Enumerate the constellation symbols based on119898119860 for
all layers
The tree-search step is as follows
(1) Set layer to 2119873119905 listL = 0 119889
2119873119905+1
= 0
(a) expand all radic2119876 possible constellation nodes(b) calculate the corresponding PEDs(c) if radic2119876 gt 119870 select the 119870 best nodes and store
them in the listL2119873119905
(2) For layer 119894 = 2119873119905minus 1 1
(a) enumerate the constellation point according to119898119862
119894of the 119870 surviving paths in the listL
119894+1
(b) find the first child (FC) based on 119898119862
119894and 119898
119860
119894for
each parent nodes(c) compute their PEDs(d) select 119860 best children with smallest PEDs
among the 119870 FCs and add them to the listL119894
(e) if |L119894| lt 119870 find the next child (NC) of the
selected parent nodesCalculate their PEDs and go to step (2)(d)
(f) else move to the next layer 119894 = 119894 minus 1 and go tostep (2)
(3) If 119894 = 1 calculate the LLR as in (19)In the case of missing counter hypothesis LLR clip-ping method is used
It has been shown in [20] that the LC-K-Best decoderachieves almost the same performance as the classical K-Best decoder with different modulations Moreover thecomputational complexity in terms of the number of visitednodes is significantly reduced specially in the case of high-order modulations
5 Convergence of IterativeDetection-Decoding
The EXtrinsic Information Transfer (EXIT) chart is a usefultool to study the convergence behavior of iterative decod-ing systems [50] It describes the exchange of the mutualinformation in the iterative process in order to predict therequired number of iterations the convergence threshold(corresponding to the start of the waterfall region) and theaverage decoding trajectory
In the iterative receiver considered in our study two itera-tive processes are performed one inside the channel decoder(turbo or LDPC) and the other between the MIMO detectorand the channel decoder For simplicity we separately studythe convergence of the channel decoding and the MIMOdetection We denote by 119868
1198601and 119868
1198602the a priorimutual input
information of the MIMO detector and the channel decoderrespectively and by 119868
1198641and 119868
1198642their corresponding extrinsic
mutual output informationThe mutual information 119868
119909(119868119860or 119868
119864) can be computed
through Monte Carlo simulation using the probability den-sity function 119901
119871119909
[50]
119868119909
=1
2sum
119909isinminus11
int
+infin
minusinfin
119901119871119909
(119871119909
| 119909)
sdot log2
2119901119871119909
(119871119909
| 119909)
119901119871119909
(119871119909
| minus1) 119901119871119909
(119871119909
| +1)119889119871
119909
(29)
A simple approximation of the mutual information is used inour analysis [51]
119868119909
asymp 1 minus1
119871119887
119871119887minus1
sum
119899=0
log2
(1 + exp (minus119909119871119909)) (30)
where 119871119887is the number of transmitted bits and 119871
119909is the LLR
associated with the bit 119909 isin minus1 +1The a priori information 119871
119860can be modeled by applying
an independentGaussian randomvariable 119899119860with zeromean
and variance 1205902
119860in conjunction with the known transmitted
information bits 119909 [50]
119871119860
= 120583119860
119909 + 119899119860
where 120583119860
=1205902
119860
2 (31)
For each given mutual information value 119868119860
isin [0 1] 1205902
119860can
be computed using the following equation [52]
120590119860
asymp (minus1
1198671
log2
(1 minus 11986811198673
119860))
121198672
(32)
where 1198671
= 03073 1198672
= 08935 and 1198673
= 11064At the beginning the a priori mutual information is as
follows 1198681198601
= 0 and 1198681198602
= 0 Then the extrinsic mutualinformation 119868
1198641of the MIMO detector becomes the a priori
mutual information 1198681198602
of the channel decoder and so onand so forth (ie 119868
1198641= 119868
1198602and 119868
1198642= 119868
1198601) For a successful
decoding there must be an open tunnel between the curves
Mobile Information Systems 9
Table 1 Simulation parameters for convergence analysis
MIMO system 4 times 4 spatial multiplexingModulation 16-QAM Gray mappingChannel Flat Rayleigh fading
DetectorSTS-SD 119871 clip = plusmn8
LC-119870-Best 119870 = 16 119871 clip = plusmn3
MMSE-IC
Channel decoder
LTE turbo code [13 15]119900
Block length 119870119887
= 1024 119877119888
= 12
LDPC code (IEEE 80211n)Codeword length 119873
119887= 1944 119877
119888= 12
Interleaver Random size = 1024 (turbo code case)
the exchange of extrinsic information can be visualized as aldquozigzagrdquo decoding trajectory in the EXIT chart
To visualize the exchange of extrinsic information of theiterative receiver we present the MIMO detector and thechannel decoder characteristics into a single chart For ourconvergence analysis a 4 times 4 MIMO system with 16-QAMconstellation turbo decoder and LDPCdecoder (119877
119888= 12) is
considered Table 1 summarizes the main parameters for theconvergence analysis
Figure 5 shows the extrinsic information transfer char-acteristics of MIMO detectors at different 119864
119887119873
0values
As the I-VBLAST detector performs successive interferencecancellation at the first iteration and parallel interference can-cellation of the soft estimated symbols for the rest iterationsit is less intuitive to present its convergence in the EXITchart Therefore the convergence analysis of VBLAST is notconsidered
It is obvious that the characteristics of the detectorsare shifted upward with the increase of 119864
119887119873
0 We show
that for low 119864119887119873
0(0 dB) and for low mutual information
(lt01) MMSE-IC performs better than LC-K-Best decoderHowever with larger mutual information its performance islower Moreover the mutual information of STS-SD is higherthan LC-K-Best decoder and MMSE-IC for different 119864
119887119873
0
For higher 119864119887119873
0(5 dB) MMSE-IC presents lower mutual
information than other decoders when 1198681198601
lt 09Figure 6 shows the EXIT chart for 119864
119887119873
0= 2 dB with
several MIMO detectors namely STS-SD LC-K-Best andMMSE-IC We note that the characteristic of the channeldecoder is independent of 119864
119887119873
0values It is obvious that
the extrinsic mutual information of the channel decoderincreases with the number of iterations We see that after 6 to8 iterations in the case of turbo decoder (Figure 6(a)) thereis no significant improvement on the mutual informationMeanwhile in the case of LDPC decoder in Figure 6(b) 20iterations are enough for LDPC decoder to converge
By comparing the characteristics of STS-SD LC-K-Bestdecoder andMMSE-IC equalizer with both coding schemeswe notice that STS-SD has a larger mutual informationat its output LC-K-Best decoder has slightly less mutualinformation than STS-SD MMSE-IC shows low mutualinformation levels at its output compared to other algorithms
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
STS-SDLC-K-Best
MMSE-IC
EbN0 = 0 2 5dB
5dB
2dB
0dBI E1I
A2
IA1 IE2
Figure 5 Extrinsic information transfer characteristics of MIMOdetectors (STS-SD LC-K-Best andMMSE-IC) at119864
119887119873
0= 0 2 5 dB
in a 4 times 4 MIMO system using 16-QAM
when 1198681198601
lt 09 while for 1198681198601
gt 09 the extrinsic mutualinformation is comparable to others
In the case of turbo decoder (Figure 6(a)) with 119868in =
8 3 outer iterations are sufficient for STS-SD to convergeat 119864
119887119873
0= 2 dB However the same performance can
be attained by performing 4 outer iterations with only 2inner iterations Similarly LC-K-Best decoder shows anequivalent performance but slightly higher119864
119887119873
0is required
The convergence speed of LC-K-Best decoder is a bit lowerthan STS-SD which requires more iterations to get the sameperformance The reason is mainly due to the unreliabilityof LLRs caused by the small list size (L = 16) In thecase of MMSE-IC the characteristic presents a lower mutualinformation than the LC-K-Best decoder when 119868
1198601lt 09
Therefore an equivalent performance can be obtained athigher 119864
119887119873
0or by performing more iterations
In a similar way we study the convergence of MIMOdetection algorithms with LDPC decoder Figure 6(b) showsthe EXIT chart at 119864
119887119873
0= 2 dBThe same conclusion can be
retrieved as in the case of turbo decoder We can see that at119864119887119873
0= 2 dB a clear tunnel is observed between the MIMO
detector and the channel decoder characteristics allowingiterations to bring improvement to the system SimilarlySTS-SD offers higher mutual information than LC-K-Bestdecoder andMMSE-IC equalizer which suggests its superiorsymbol detection performance
Additionally the average decoding trajectory resultingfrom the free-run iterative detection-decoding simulationsis illustrated in Figure 6 at 119864
119887119873
0= 2 dB with 119868out = 4
119868in = 2 and 119868out = 4 119868in = 5 in the case of turbo decoder andLDPC decoder respectively The decoding trajectory closely
10 Mobile Information Systems
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
TurboSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 2 6 8
(a)
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
LDPCSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 5 10 20
(b)
Figure 6 EXIT chart for MIMO detectors (STS-SD LC-K-Best and MMSE-IC) and channel decoder (a) LTE turbo decoder 119870119887
= 1024119877119888
= 12 and (b) LDPC decoder 119873119887
= 1944 119877119888
= 12 at 119864119887119873
0= 2 dB in a 4 times 4 MIMO system using 16-QAM
matches the characteristics in the case of STS-SD and LC-K-Best decoders The little difference from the characteristicsafter a few iterations is due to the correlation of extrinsicinformation caused by the limited interleaver depth In thecase of MMSE-IC the decoding trajectory diverges fromthe characteristics for high mutual information because theequalizer uses the a posteriori information to compute softsymbols instead of the extrinsic information
The best trade-off scheduling of the required numberof iterations is therefore 119868out iterations in the outer loopand a total of 8 iterations inside the turbo decoder and 20iterations inside the LDPC decoder distributed across these119868out iterations
6 Performance and Complexity Evaluation ofIterative Detection-Decoding
In this section we evaluate and compare the performance andthe complexity of differentMIMOdetectors namely STS-SDLC-K-Best decoder MMSE-IC and I-VBLAST equalizerswith different channel coding techniques (turbo LDPC)A detailed analysis of the performance and the complexitytrade-off ofMIMOdetection with LTE turbo decoder and 16-QAM modulation in a Rayleigh channel has been discussedin [24] Herein the performance and the complexity ofthe receiver with LDPC decoder are investigated Moreoverseveral modulations and coding schemes are consideredto quantify the gain achieved by such iterative receiver indifferent channel environments Consequently a comparative
study is conducted in iterative receiver with both codingschemes (turbo LDPC)
For the turbo code the rate 13 turbo encoder specifiedin 3GPP LTE with a block length 119870
119887= 1024 is used in the
simulations Puncturing is performed in the rate matchingmodule to achieve an arbitrary coding rate 119877
119888(eg 119877
119888= 12
34)Meanwhile the LDPC encoder specified in IEEE 80211nis consideredThe encoder is defined by a parity checkmatrixthat is formed out of square submatrices of sizes 27 54 or 81Herein the codeword length of size 119873
119887= 1944 with coding
rate of 119877119888
= 12 and 34 is considered
61 Performance Evaluation The simulations are first carriedout in Rayleigh fading channel to view general performanceof the iterative receivers Real channel models will be consid-ered to evaluate the performance in more realistic scenariosTherefore the 3GPP LTE-(A) channel environments withlow medium and large delay spread values and Dopplerfrequencies are considered The low spread channel is theExtended Pedestrian A (EPA) model which emulates theurban environment with small cell sizes (120591rms = 43 ns) Themedium spread channel (120591rms = 357 ns) is the ExtendedVehicular A (EVA) model The Extended Typical Urban(ETU) model is the large spread channel which has alarger excess delay (120591rms = 991 ns) and simulates extremeurban suburban and rural cases Table 2 summarizes thecharacteristic parameters of these channel environments Forall cases the channel is assumed to be perfectly known atthe receiver Table 3 gives the principle parameters of thesimulations The performance is measured in terms of bit
Mobile Information Systems 11
Table 2 Characteristic parameters of the investigated channelmodels
error rate (BER) with respect to signal-to-noise ratio (SNR)per bit 119864
119887119873
0
119864119887
1198730
=119864119904
1198730
+ 10 sdot log10
1
119877119888119876119873
119905
[dB] (33)
In our previous study [24] the performance of MIMOdetectors with LTE turbo decoder is evaluated in a Rayleighchannel with various outer and inner iterations It has beenshown that the performance is improved by about 15 dB at aBER level of 1 times 10
minus5 with 4 outer iterations It has been alsoshown that no significant improvement can be achieved after4 outer iterations this improvement is less than 02 dB with 8outer iterations
Similarly to turbo decoder we fix the number of inneriterations inside LDPC decoder to 20 while varying thenumber of outer iterations Figure 7 shows BER performanceof MIMO detectors with LDPC decoder in Rayleigh channelwith 119868in = 20 iterations and 119868out = 1 2 4 or 8 iterationsSTS-SD is used with a LLR clipping level of 8 which givesclose toMAPperformancewith considerable reduction in thecomplexity For LC-K-Best decoder a LLR clipping level of 3is used in the case of missing counter hypothesis We show
0 1 2 3 4 5
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
Iout = 4
Iout = 8
Iout = 2
Iout = 1
EbN0 (dB)minus1
Figure 7 BER performance of a 4 times 4 spatial multiplexing systemwith 16-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) in a Rayleigh fading channel 119868out = 1 2 4 8 119868in =
20 LDPC decoder 119873119887
= 1944 and 119877119888
= 12
that performance improvement of 15 dB is observed with 4outer iterations For 119868out = 8 iterations the improvementis less than 02 dB Therefore 119868out = 4 iterations will beconsidered in the sequel
Figure 8 shows the BER performance of 4 times 4 16-QAMsystem in a Rayleigh fading channel with 119868out = 4 and 119868in = 2
in the case of turbo decoder and 119868in = [3 4 6 7] in the caseof LDPC decoder The notation 119868in = [3 4 6 7] denotes that3 inner iterations are performed in the 1st outer iteration 4inner iterations in the 2nd outer iteration and so on Theperformances of STS-SD with 119868in = 8 and 119868in = 20 foreach outer iteration in the case of turbo decoder and LDPCdecoder are also plotted as a reference In the case of turbodecoder we show that performing 119868in = 8 and 119868out = 4
iterations does not bring significant improvement on theperformance compared to the case when 119868in = 2 and 119868out =
4 iterations are performed Similarly in the case of LDPCdecoder the performance of 119868in = 20 and 119868out = 4 iterationsis comparable to the performance of 119868in = [3 4 6 7] and119868out = 4 iterations Hence using a large number of iterationsdoes not seem to be efficientwhich proves the results obtainedin the convergence analysis of Section 5
By comparing the algorithms LC-K-Best decoder showsa degradation of less than 02 dB compared to STS-SD ata BER level of 2 times 10
minus5 However it outperforms MMSE-IC and I-VBLAST equalizer by about 02 dB at a BERlevel of 2 times 10
minus5 MMSE-IC and I-VBLAST show almostthe same performance In addition in the case of LDPCdecoder (Figure 8(b)) we notice that increasing the numberof inner iterations for each outer iteration 119868in = [3 4 6 7]
12 Mobile Information Systems
0 05 1 15 2 25 3
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 8
(a) LTE turbo decoder
BER
0 05 1 15 2 25 3
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 20
STS-SD Iin = 5
STS-SD
LC-K-BestI-VBLASTMMSE-IC
(b) LDPC decoder
Figure 8 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
shows slightly better performance than performing an equalnumber of iterations 119868in = 5 for each outer iteration
In the case of high-order modulation higher spectralefficiency can be achieved at a cost of increased symboldetection difficulty Figure 9 shows the BER performance of64-QAM with 119877
119888= 34 We see that LC-K-Best decoder
with a list size of 32 presents the similar performance as STS-SD However I-VLAST equalizer and MMSE-IC equalizerpresent degradation of more than 2 dB at a BER level of 1 times
10minus5 compared to LC-K-Best decoder Therefore LC-K-Best
decoder is more robust in the case of high-ordermodulationsand high coding rates The figure also shows that the BERperformance of LDPC decoder is almost identical to that ofthe turbo decoder
In order to summarize the performance of differentdetectors with different channel decoders we provide the119864119887119873
0values achieving a BER level of 1 times 10
minus5 in Table 4The values given in the parentheses of the table represent theperformance loss compared to STS-SD
Next we evaluate the performance of the iterative receiverin more realistic channel environments Figures 10 11 and 12show the BER performance of the detectors with the channeldecoders on EPA EVA and ETU channels receptivelySimilar behaviors can be observed with LTE turbo decoderand with LDPC decoder
In EPA channel (Figure 10) we see that LC-K-Bestdecoder achieves similar performance as STS-SD in the caseof 64-AQM and presents a degradation less than 02 dB in the
case of 16-QAM Meanwhile MMSE-IC presents significantperformance loss of more than 6 dB in the case of 64-QAMand 119877
119888= 34 With 16-QAM and 119877
119888= 12 the degradation
of MMSE-IC compared to LC-K-Best decoder is about 1 dBat a BER level of 1 times 10
minus4In EVA channel (Figure 11) the performance loss of
MMSE-IC compared to LC-K-Best decoder is reduced toapproximately 5 dB with 64-QAM and 05 dB with 16-QAMLC-K-Best decoder presents a degradation of about 01sim03 dB compared to STS-SD in the case of 16-QAM and 64-QAM
Similarly in ETU channel (Figure 12) MMSE-IC presentsa performance degradation compared to LC-K-Best decoderThis degradation is less than 4 dB in the case of 64-QAM andless than 05 dB in the case of 16-QAM We notice also thatthe LC-K-Best decoder is comparable to STS-SD in the caseof 64-QAM and has a degradation of 02 dB in the case of 16-QAM
Comparing the performance of the iterative receiver indifferent channels it can be seen that iterative receiverspresent the best performance in ETU channel compared toEPA and EVA channels This is due to the high diversity ofETU channel At a BER level of 1 times 10
minus4 the performancegain in ETU channel in the case of LTE turbo decoder is about08 dB 13 dB compared to EVA channel with 16-QAM and64-QAM respectively In the case of LDPC decoder this gainis 04 dB and 1 dB with 16-QAM and 64-QAM respectivelyHowever in EPA channel the performance gain in ETU
Mobile Information Systems 13
3 4 5 6 7 8 9 10 11 12 13
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
EbN0 (dB)
(a) LTE turbo decoder
3 4 5 6 7 8 9 10 11 12
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)
STS-SD Iin = 20
STS-SDLC-K-Best
I-VBLASTMMSE-IC
(b) LDPC decoder
Figure 9 BER performance of a 4 times 4 spatial multiplexing system with 64-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
0values achieving a BER level of 1times 10minus5 for different detectors and channel decoders (turbo LDPC) in 4times 4 spatialmultiplexing
system with 16-QAM 119877119888
= 12 and 64-QAM 119877119888
= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
STS-SD 25 dB 102 dB 27 dB 96 dBLC-119870-Best 26 dB (minus01) 102 dB (00) 28 dB (minus01) 96 dB (00)I-VBLAST 28 dB (minus03) 128 dB (minus26) 29 dB (minus02) 12 dB (minus24)MMSE-IC 28 dB (minus03) 128 dB (minus26) 30 dB (minus03) 12 dB (minus24)lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
channel in the case of turbo decoder or LDPC decoder ismore than 1 dB with 16-QAM and 64-QAM
Table 5 summarizes the 119864119887119873
0values achieving a BER
level of 1times10minus4 of different detectors combined with different
channel decoders andmodulation orders in different channelmodels The values given in the parentheses in the tablerepresent the performance loss compared to STS-SD Asindicated in Table 5 the iterative receivers with turbo decoderand LDPC decoder have comparable performance with acoding rate 119877
119888= 12 (16-QAM) However with 119877
119888= 34
(64-QAM) the receivers with LDPC decoder present slightlya better performance especially in ETU channel (06 dB)
From these results we show that the iterative receiversubstantially improves the performance of coded MIMOsystems either with turbo decoder or with LDPC decoderin Rayleigh channel (Figures 8 and 9) and in more realisticchannels (Figures 10 11 and 12) Moreover we show that
performing a large number of inner iterations does notbring significant improvement In addition we show thatthe LC-K-Best decoder achieves a good performance withdifferent modulations and channel coding schemes Thefigures suggest that the BER performance of the iterativereceiver with turbo decoder is almost comparable to that ofthe LDPC decoder It is therefore meaningful to evaluate thecomputational complexity of the iterative receivers with bothdecoding techniques as it will be discussed in the next section
62 Complexity Evaluation The computational complexityhas significant impact on the latency throughput and powerconsumption of the deviceTherefore the receiver algorithmsshould be optimized to achieve a good trade-off betweenperformance and cost In this part we evaluate the com-putational complexity of the iterative receiver in terms ofbasic operations such as addition subtractionmultiplication
14 Mobile Information Systems
1 3 5 7 9 11 13 15 17 19
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
(a) EPA turbo 119868out = 4 119868in = 2
BER
1 3 5 7 9 11 13 15 17 19
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12 64-QAM
Rc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EPA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 10 BER performance of a 4 times 4 spatial multiplexing with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) on EPA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944 119868out = 4
119868in = [3 4 6 7]
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
STS-SDLC-K-Best
MMSE-IC
16-QAMRc = 12
64-QAMRc = 34
(a) EVA turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EVA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 11 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Best and MMSE-IC) on EVA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944
and 119868out = 4 119868in = [3 4 6 7]
Mobile Information Systems 15
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(a) ETU turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) ETU LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 12 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SDLC-K-Best and MMSE-IC) on ETU channel model (a) LTE turbo decoder 119870
minus4 in LTE channel models for different detectors and channel decoders (turbo LDPC)in 4 times 4 spatial multiplexing system with 16-QAM 119877
119888= 12 and 64-QAM 119877
119888= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
EPASTS-SD 63 dB 140 dB 62 dB 138 dB
LC-119870-Best 65 dB (minus02) 139 dB (+01) 64 dB (minus02) 139 dB (minus01)MMSE-IC 74 dB (minus11) gt20 dB (gtminus6) 80 dB (minus18) gt20 dB (gtminus6)
EVASTS-SD 52 dB 143 dB 53 dB 134 dB
LC-119870-Best 54 dB (minus02) 144 dB (minus01) 56 dB (minus03) 137 dB (minus03)MMSE-IC 58 dB (minus06) 190 dB (minus47) 60 dB (minus07) 185 dB (minus51)
ETUSTS-SD 44 dB 130 dB 49 dB 124 dB
LC-119870-Best 46 dB (minus02) 130 dB (00) 49 dB (00) 124 dB (00)MMSE-IC 49 dB (minus05) 175 dB (minus35) 51 dB (minus02) 16 dB (minus36)
lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
division square root extraction maximization and look-up table check (which are denoted by ADD SUB MULDIV SQRT Max and LUT resp) To this end the completecomparison of the iterative receivers with both channeldecoders (turbo LDPC) and with several modulations andcoding rates is carried out
621 Complexity of Iterative Receiver The complexity ofiterative receiver depends on theMIMOdetector the channeldecoder and the number of innerouter iterationsThis com-plexity can be expressed by
sdot [119862det1 + (119868out minus 1) sdot 119862deti] (34)
where 119862det1 denotes the complexity of the first iterationof MIMO detection algorithm per symbol vector withouttaking into consideration the a priori information 119862detidenotes the complexity per iteration per symbol vector takinginto consideration the a priori information 119862dec denotesthe complexity of the channel decoder per iteration perinformation bit 119873bit is the number of information bit at theinput of the encoder 119873symb is the number of symbol vectors119873symb and 119873bit are linked by the following relation
119873symb =119873bit
119876119877119888119873119905
= 120572119873bit with 120572 =1
119876119877119888119873119905
(35)
where 119876 is the number of bits in the constellation symbol 119877119888
is the coding rate and119873119905is the number of transmit antennas
16 Mobile Information Systems
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
Figure 2 Structure of LTE turbo code with 119877119888
= 13 (a) turboencoder and (b) turbo decoder
compute the extrinsic information as shown in Figure 2(b)Two families of decoding algorithms can be used soft-output Viterbi algorithms (SOVA) [27 28] and maximum aposteriori (MAP) algorithm [29] The MAP algorithm offerssuperior performance but suffers from high computationalcomplexity Two suboptimal algorithms namely log-MAPand max-log-MAP are practically used [30] Herein log-MAPalgorithm is considered by using the Jacobian logarithm[30]
where 119891119888(|119909 minus 119910|) is a correction function that can be
computed using a small look-up table (LUT)The decoder computes the branch metrics (120574) and the
forward (120572) and the backward (120573) metrics between two statesin the trellis as follows
120574119896
(119904119896minus1
119904119896) = 119901 (119904
119896 119910119896
| 119904119896minus1
)
120572119896
(119904119896) =
lowastmax119904119896minus1
(120572119896minus1
(119904119896minus1
) + 120574119896
(119904119896minus1
119904119896))
120573119896
(119904119896) =
lowastmax119904119896+1
(120573119896+1
(119904119896+1
) + 120574119896
(119904119896 119904119896+1
))
(5)
The a posteriori LLRs of the information bits are computed as
119871 (119906119896) =
lowastmax119906119896=0
(120572119896minus1
(119904119896minus1
) + 120574119896
(119904119896minus1
119904119896) + 120573
119896(119904119896))
minuslowastmax119906119896=1
(120572119896minus1
(119904119896minus1
) + 120574119896
(119904119896minus1
119904119896) + 120573
119896(119904119896))
(6)
The component decoders exchange only the extrinsic LLRwhich is defined by
119871119890(119906119896) = 119871 (119906
119896) minus 119871
119886(119906119896) minus 119871
119904(119906119896) (7)
where 119871119886(119906119896) and 119871
119904(119906119896) correspond to the a priori informa-
tion from the other decoder and the systematic informationbits respectively
32 LDPC Decoder LDPC codes belong to a class of linearerror correcting block codes first proposed by Gallager [26]Their main advantages lie in their capacity-approaching per-formance and their low-complexity parallel implementations[31] LDPC codes can be represented by a parity checkmatrix HLDPC or intuitively through Tanner graph [32]Tanner graphs are bipartite graphs containing two types ofnodes the check nodes and the variable nodes as illustratedin Figure 3(a) It consists of 119872 check nodes (CN) whichcorrespond to the number of parity bits (ie number of rowsof HLDPC) and 119873 variable nodes (VN) corresponding to thenumber of bits in a codeword (ie number of columns ofHLDPC)
The optimal maximum a posteriori decoding of LDPCcodes is infeasible from the practical implementation point ofviewAlternatively LDPCdecoding is done using themessagepassing or belief propagation algorithms which iterativelypass messages between check nodes and variable nodes asshown in Figure 3(b) The belief propagation is denotedas the sum-product decoding because probabilities can berepresented as LLRs which allow the calculation of messagesusing sum and product operations
Let 119871V119894119895
be themessage from variable node 119894 to check node119895 and 119871
119888119895119894
the message from check node 119895 to variable node119894 Let 119881
119895and 119862
119894denote the set of adjacent variable nodes
connected to the check node 119895 and the set of adjacent checknodes connected to the variable node 119894 respectively where0 le 119895 le 119872 and 0 le 119894 le 119873 For the first iteration theinput to the LDPC decoder is the LLRs 119871(119888
119894) of the codeword
c which are used as an initial value of the extrinsic variablenode messages that is 119871V
119894119895
= 119871(119888119894) For the 119896th iteration the
algorithm can be summarized as follows(1) Each check node 119895 computes the extrinsic message to
its neighboring variable node 119894
119871119888119895119894
= 2tanhminus1 ( prod
1198941015840isin119881119895119894
tanh(
119871V1198941015840119895
2)) (8)
(2) Each variable node 119894 updates its extrinsic informationto the check node 119895 in the next iteration
119871V119894119895
= 119871 (119888119894) + sum
1198951015840isin119862119894119895
1198711198881198951015840119894
(9)
Mobile Information Systems 5
n = 6m = 4
VN (n)
CN (m)
0 1 1 0 0 1
1 1 1 0 1 0
1 0 0 1 1 1
0 0 1 1 0 1
=HLDPC
(a)
1 2 3 5 64
1 2 3 4
VNi
CNj
L13
L12
L31
L32
L34
Lc12Lc13
Lc16Lc31
Lc34
Lc35
Lc36
(b)
Figure 3 LDPC decoder (a) matrix representation and Tanner graph and (b) iterative decoding between VNs and CNs
(3) The a posterioriLLRof each codeword bit is computedas
119871119901
(119888119894) = 119871 (119888
119894) + sum
1198951015840isin119862119894
1198711198881198951015840119894
(10)
The decoding algorithm alternates between check node proc-essing and variable node processing until a maximum num-ber of iterations are achieved or until the parity check con-dition is satisfied When the decoding process is terminatedthe decoder outputs the a posteriori LLR
4 Soft-Input Soft-Output MIMO Detection
The aim of MIMO detection is to recover the transmittedvector s from the received vector y The state-of-the-artMIMO detection algorithms have been presented in [24]These algorithms can be divided into two main familiesnamely the tree-search-based detection and the interference-cancellation-based detection In this section we brieflyreview the main existing SISO MIMO detection algorithmsuseful for the following sections
41 MaximumA Posteriori Probability (MAP) Detection TheMAP algorithm achieves the optimum performance throughthe use of an exhaustive search over all 2
119876sdot119873119905 possible symbol
combinations to compute the LLR of each bit The LLR of the119887th bit in the 119894th transmit symbol 119909
119894119887 is given by
119871 (119909119894119887
) = log119875 (119909
119894119887= +1 | y)
119875 (119909119894119887
= minus1 | y)
= logsumsisin120594+1
119894119887
119901 (y | s) 119875 (s)
sumsisin120594minus1119894119887
119901 (y | s) 119875 (s)
(11)
where 120594+1
119894119887and 120594
minus1
119894119887denote the sets of symbol vectors in
which the 119887th bit in the 119894th antenna is equal to +1 and minus1respectively 119901(y | s) is the conditioned probability densityfunction given by
119901 (y | s) =1
(1205871198730)119873119903
exp(minus1
1198730
1003817100381710038171003817y minus Hs10038171003817100381710038172
) (12)
119875(s) represents the a priori information provided by thechannel decoder in the form of a priori LLRs
119871119860
(119909119894119887
) = log119875 (119909
119894119887= +1)
119875 (119909119894119887
= minus1) forall119894 119887
119875 (s) =
119873119905
prod
119894=1
119875 (119904119894) =
119873119905
prod
119894=1
119876
prod
119887=1
119875 (119909119894119887
)
(13)
The max-log-MAP approximation is commonly used in theLLR calculation with lower complexity [12]
119871 (119909119894119887
) asymp1
1198730
min120594minus1
119894119887
1198891 minus
1
1198730
min120594+1
119894119887
1198891 (14)
1198891
=1003817100381710038171003817y minus Hs1003817100381710038171003817
2
minus 1198730
119873119905
sum
119894=1
119876
sum
119887=1
log119875 (119909119894119887
) (15)
where 1198891represents the Euclidean distance between the
received vector y and lattice pointsHsBased on the a posteriori LLRs 119871(119909
119894119887) and the a priori
LLRs 119871119860
(119909119894119887
) the detector computes the extrinsic LLRs119871119864(119909119894119887
) as
119871119864
(119909119894119887
) = 119871 (119909119894119887
) minus 119871119860
(119909119894119887
) (16)
The MAP algorithm is not feasible due to its exponentialcomplexity since 2
119876sdot119873119905 hypotheses have to be considered
within each minimum term and for each bit Thereforeseveral suboptimal MIMO detectors have been proposedwith reduced complexity as will be briefly discussed in thefollowing sections
42 Tree-Search-Based Detection The tree-search-baseddetection methods generally fall into two main categoriesnamely depth-first search like the sphere decoder andbreadth-first search like the K-Best decoder
421 List Sphere Decoder (LSD) The basic idea of the spheredecoder is to limit the search space of the MAP solution to ahypersphere of radius 119903
119904around the received vector Instead
of testing all the hypotheses of the transmitted signal onlythe lattice points that lie inside the hypersphere are testedreducing the computational complexity [33]
sSD = arg minsisin2119876119873119905
1003817100381710038171003817y minus Hs1003817100381710038171003817
2
le 1199032
119904 (17)
6 Mobile Information Systems
Using the QR decomposition in real-valued model thechannel matrix H can be decomposed into two matrixes Qand R (H = QR) where Q is 2119873
119903times 2119873
119905orthogonal matrix
(Q119867Q = I2119873119905
) and R is 2119873119905
times 2119873119905upper triangular matrix
with real-positive diagonal elements [34] Therefore the dis-tance in (17) can be computed as yminusHs2 = yminusRs2 wherey = Q119867y is the modified received symbol vector Exploitingthe triangular nature of R the Euclidean distance metric 119889
1
in (15) can be recursively evaluated through the accumulatedpartial Euclidean distance (PED) 119889
and the a priori-based partial metric at the 119894th level respec-tively
This process can be illustrated by a tree with 2119873119905
+ 1
levels as depicted in Figure 4(a) The tree search starts at theroot level with the first child node at level 2119873
119905 The partial
Euclidean distance 1198892119873119905
in (18) is then computed If 1198892119873119905
issmaller than the sphere radius 119903
119904 the search continues at level
2119873119905minus1 and steps down the tree until finding a valid leaf node
at level 1List sphere decoder is proposed to approximate the MAP
detector [12] It generates a list L sub 2119876119873119905 that includes the
best possible hypotheses The LLR values are then computedfrom this list as
119871 (119909119894119887
) =1
1198730
minLcap120594minus1119894119887
1198891 minus
1
1198730
minLcap120594+1119894119887
1198891 (19)
The main issue of LSD is the missing counter-hypothesisproblem depending on the list sizeThe use of limited list sizecauses inaccurate approximation of the LLR due to missingsome counter hypotheses where no entry can be found inthe list for a particular bit 119909
119894119887= +1 minus1 Several solutions
have been proposed to handle this issue LLR clipping is afrequently used solution which consists simply to set the LLRto a predefined maximum value [12 35]
Several methods can be considered to reduce the com-plexity of the sphere decoder such as Schnorr-Euchner(SE) enumeration [36] layer ordering technique [34] andchannel regularization [37] Layer ordering technique allowsthe selection of the most reliable symbols at a high layerusing the sorted QR (SQR) decomposition However channelregularization introduces a biasing factor in the metricswhich should be removed in LLR computation to avoidperformance degradation as discussed in [38] In the sequelthe SQR decomposition is considered in the preprocessingstep
422 Single Tree-Search SphereDecoder (STS-SD) Oneof thetwominima in (14) corresponds to theMAPhypothesis sMAPwhile the other corresponds to the counter hypothesis Thecomputation of LLR can be expressed by
119871 (119909119894119887
) =
1
1198730
(119889MAP119894119887
minus 119889MAP
) if 119909MAP119894119887
= +1
1
1198730
(119889MAP
minus 119889MAP119894119887
) if 119909MAP119894119887
= minus1
(20)
with
119889MAP
=10038171003817100381710038171003817y minus RsMAP10038171003817100381710038171003817
2
minus 1198730119875 (sMAP
)
119889MAP119894119887
= min119904isin120594
MAP119894119887
1003817100381710038171003817y minus Rs1003817100381710038171003817
2
minus 1198730119875 (s)
sMAP= arg min
119904isin2119876sdot119872
1003817100381710038171003817y minus Rs1003817100381710038171003817
2
minus 1198730119875 (s)
(21)
where 120594MAP119894119887
denotes the bitwise counter hypothesis of theMAP hypothesis which is obtained by searching over all thesolutions with the 119887th bit of the 119894th symbol opposite to thecurrentMAPhypothesisOriginally theMAPhypothesis andthe counter hypotheses can be found through repeating thetree search [39] The repeated tree search yields a large com-putational complexity cost To overcome this the single tree-search algorithm [13 40] was developed to compute all theLLRs concurrently The 119889
MAP metric and the corresponding119889MAP119894119887
metrics are updated through one tree-search processThrough the use of extrinsic LLR clipping method the STS-SD algorithm can be tunable between the MAP performanceand hard-output performance The implementations of STS-SD have been reported in [14 15]
423 SISO K-Best Decoder K-Best algorithm is a breadth-first search based algorithm in which the tree is traversedonly in the forward direction [41] This approach searchesonly a fixed number 119870 of paths with best metrics at eachdetection layer Figure 4(b) shows an example of the treesearch with 119873
119905= 2 The algorithm starts by extending the
root node to all possible candidates It then sorts the newpaths according to their metrics and retains the 119870 paths withsmallest metrics for the next detection layer
K-Best algorithm is able to achieve near-optimal perfor-mance with a fixed and affordable complexity for parallelimplementation Yet the major drawbacks ofK-Best decoderare the expansion and the sorting operations that are verytime consuming Several proposals have been drawn in theliterature to approximate the sorting operations such asrelaxed sorting [42] and distributed sorting [43] or evento avoid sorting using on-demand expansion scheme [44]Moreover similarly as LSD K-Best decoder suffers frommissing counter-hypothesis problem due to the limited listsize Numerous approaches have been proposed to addressthis problem such as smart candidates adding [45] bit flip-ping [46] and path augmentation and LLR clipping [12 35]
43 Interference-Cancellation-Based Detection Interference-cancellation-based detection can be carried out either in aparallel way as inMMSE-IC [8 9] or in a successive way as inVBLAST [47]
431 Minimum Mean Square Error-Interference Cancella-tion (MMSE-IC) Equalizer MMSE-IC equalizer can be per-formed using two filters [4]The first filter p
119894is applied to the
received vector y and the second filter q119894is applied to the
estimated vector s in order to cancel the interference fromother layers The equalized symbol
119894can be written as
119894= p119867
119894y minus q119867
119894s119894
with 119894 isin [1 119873119905] (22)
where s119894denotes the estimated vector given by the
previous iteration with the 119894th symbol omitteds119894
= [1 sdot sdot sdot 119894minus1
0 119894+1
sdot sdot sdot 119873119905]
119894is calculated by the
soft mapper as 119894
= E[119904119894] = sum
119904isin2119876 119904119875 (119904
119894= 119904) [48] The filters
p119894and q
119894are optimized using the MMSE criterion and are
given in [6 24]For the first iteration since no a priori information is
available the equalization process is reduced to the classicalMMSE solution
119894= (H119867H +
1205902
119899
1205902119904
I119873119905
)
minus1
H119867y (23)
The equalized symbols 119894are associated with a bias factor 120573
119894
in addition to some residual noise plus interferences 120578119894
119894= 120573
119894119904119894+ 120578
119894 (24)
These equalized symbols are then used by the soft demapperto compute the LLR values using the max-log-MAP approxi-mation [48]
119871 (119909119894119887
) =1
1205902120578119894
(min119904119894isin120594minus1
119894119887
1003816100381610038161003816119894 minus 120573119894119904119894
1003816100381610038161003816
2
minus min119904119894isin120594+1
119894119887
1003816100381610038161003816119894 minus 120573119894119904119894
1003816100381610038161003816
2
) (25)
MMSE-IC equalizer requires 119873119905matrix inversions for each
symbol vector For this reason several approximations of
MMSE-IC were proposed For example in [9] a low-complexity approach of MMSE-IC is described by perform-ing a single matrix inversion without performance loss Thisalgorithm is referred to as LC-MMSE-IC
432 Successive Interference Cancellation (SIC) EqualizerThe SIC-based detector was initially used in the VBLASTsystems In VBLAST architecture [47] a successive cancel-lation step followed by an interference nulling step is usedto detect the transmitted symbols However this methodsuffers from error propagation An improved VBLAST foriterative detection and decoding is described in [49] At thefirst iteration an enhanced VBLAST which takes decisionerrors into account is employed [24] When the a prioriLLRs are available from the channel decoder soft symbols arecomputed by a soft mapper and are used in the interferencecancellation To describe the enhanced VBLAST algorithmwe assume that the detection order has been made accordingto the optimal detection order [47] For the 119894th step thepredetected symbol vector s
119894minus1until step 119894 minus 1 is canceled out
from the received signal
y119894= y minus H
1119894minus1s119894minus1
(26)
where s119894minus1
= [1
2
sdot sdot sdot 119894minus1
] and H1119894minus1
=
[h1h2
sdot sdot sdot h119894minus1
] with h119894being the 119894th column of H
Then the estimated symbol 119894is obtained using a filtered
matrix W119894based on the MMSE criterion that takes decision
errors into account [11 49]
119894= W119867
119894y119894= 120573
119894119904119894+ 120578
119894
W119894= 120590
2
119904(HΣ
119894H119867 + 119873
0I119873119903
)minus1
h119894
(27)
Σ119894is the decision error covariance matrix defined as
Σ119894=
119894minus1
sum
119895=1
1205982
119895e119895e119879119895
+
119873119905minus119894+1
sum
119895=119894
1205902
119904e119895e119879119895
1205982
119895= E
10038161003816100381610038161003816119904119895
minus 119895
10038161003816100381610038161003816
2
| 119895minus1
(28)
8 Mobile Information Systems
where e119894denotes a unit vector having zero components except
the 119894th component which is oneA soft demapper is then used to compute LLRs according
to (25) We refer to this algorithm as improved VBLAST (I-VBLAST) in the sequel
44 Low-Complexity K-Best Decoder The low-complexityK-Best (LC-K-Best) decoder recently proposed in [20] usestwo improvements over the classical K-Best decoder for thesake of lower complexity and latency The first improvementsimplifies the hybrid enumeration of the constellation pointsin real-valued systemmodel when the a priori information isincorporated into the tree search using two look-up tablesThe second improvement is to use a relaxed on-demandexpansion that reduces the need of exhaustive expansionand sorting operations The LC-K-Best algorithm can bedescribed as follows
The preprocessing step is as follows
(1) Input yH 119870CalculateH = QR y = Q119867y
(2) Enumerate the constellation symbols based on119898119860 for
all layers
The tree-search step is as follows
(1) Set layer to 2119873119905 listL = 0 119889
2119873119905+1
= 0
(a) expand all radic2119876 possible constellation nodes(b) calculate the corresponding PEDs(c) if radic2119876 gt 119870 select the 119870 best nodes and store
them in the listL2119873119905
(2) For layer 119894 = 2119873119905minus 1 1
(a) enumerate the constellation point according to119898119862
119894of the 119870 surviving paths in the listL
119894+1
(b) find the first child (FC) based on 119898119862
119894and 119898
119860
119894for
each parent nodes(c) compute their PEDs(d) select 119860 best children with smallest PEDs
among the 119870 FCs and add them to the listL119894
(e) if |L119894| lt 119870 find the next child (NC) of the
selected parent nodesCalculate their PEDs and go to step (2)(d)
(f) else move to the next layer 119894 = 119894 minus 1 and go tostep (2)
(3) If 119894 = 1 calculate the LLR as in (19)In the case of missing counter hypothesis LLR clip-ping method is used
It has been shown in [20] that the LC-K-Best decoderachieves almost the same performance as the classical K-Best decoder with different modulations Moreover thecomputational complexity in terms of the number of visitednodes is significantly reduced specially in the case of high-order modulations
5 Convergence of IterativeDetection-Decoding
The EXtrinsic Information Transfer (EXIT) chart is a usefultool to study the convergence behavior of iterative decod-ing systems [50] It describes the exchange of the mutualinformation in the iterative process in order to predict therequired number of iterations the convergence threshold(corresponding to the start of the waterfall region) and theaverage decoding trajectory
In the iterative receiver considered in our study two itera-tive processes are performed one inside the channel decoder(turbo or LDPC) and the other between the MIMO detectorand the channel decoder For simplicity we separately studythe convergence of the channel decoding and the MIMOdetection We denote by 119868
1198601and 119868
1198602the a priorimutual input
information of the MIMO detector and the channel decoderrespectively and by 119868
1198641and 119868
1198642their corresponding extrinsic
mutual output informationThe mutual information 119868
119909(119868119860or 119868
119864) can be computed
through Monte Carlo simulation using the probability den-sity function 119901
119871119909
[50]
119868119909
=1
2sum
119909isinminus11
int
+infin
minusinfin
119901119871119909
(119871119909
| 119909)
sdot log2
2119901119871119909
(119871119909
| 119909)
119901119871119909
(119871119909
| minus1) 119901119871119909
(119871119909
| +1)119889119871
119909
(29)
A simple approximation of the mutual information is used inour analysis [51]
119868119909
asymp 1 minus1
119871119887
119871119887minus1
sum
119899=0
log2
(1 + exp (minus119909119871119909)) (30)
where 119871119887is the number of transmitted bits and 119871
119909is the LLR
associated with the bit 119909 isin minus1 +1The a priori information 119871
119860can be modeled by applying
an independentGaussian randomvariable 119899119860with zeromean
and variance 1205902
119860in conjunction with the known transmitted
information bits 119909 [50]
119871119860
= 120583119860
119909 + 119899119860
where 120583119860
=1205902
119860
2 (31)
For each given mutual information value 119868119860
isin [0 1] 1205902
119860can
be computed using the following equation [52]
120590119860
asymp (minus1
1198671
log2
(1 minus 11986811198673
119860))
121198672
(32)
where 1198671
= 03073 1198672
= 08935 and 1198673
= 11064At the beginning the a priori mutual information is as
follows 1198681198601
= 0 and 1198681198602
= 0 Then the extrinsic mutualinformation 119868
1198641of the MIMO detector becomes the a priori
mutual information 1198681198602
of the channel decoder and so onand so forth (ie 119868
1198641= 119868
1198602and 119868
1198642= 119868
1198601) For a successful
decoding there must be an open tunnel between the curves
Mobile Information Systems 9
Table 1 Simulation parameters for convergence analysis
MIMO system 4 times 4 spatial multiplexingModulation 16-QAM Gray mappingChannel Flat Rayleigh fading
DetectorSTS-SD 119871 clip = plusmn8
LC-119870-Best 119870 = 16 119871 clip = plusmn3
MMSE-IC
Channel decoder
LTE turbo code [13 15]119900
Block length 119870119887
= 1024 119877119888
= 12
LDPC code (IEEE 80211n)Codeword length 119873
119887= 1944 119877
119888= 12
Interleaver Random size = 1024 (turbo code case)
the exchange of extrinsic information can be visualized as aldquozigzagrdquo decoding trajectory in the EXIT chart
To visualize the exchange of extrinsic information of theiterative receiver we present the MIMO detector and thechannel decoder characteristics into a single chart For ourconvergence analysis a 4 times 4 MIMO system with 16-QAMconstellation turbo decoder and LDPCdecoder (119877
119888= 12) is
considered Table 1 summarizes the main parameters for theconvergence analysis
Figure 5 shows the extrinsic information transfer char-acteristics of MIMO detectors at different 119864
119887119873
0values
As the I-VBLAST detector performs successive interferencecancellation at the first iteration and parallel interference can-cellation of the soft estimated symbols for the rest iterationsit is less intuitive to present its convergence in the EXITchart Therefore the convergence analysis of VBLAST is notconsidered
It is obvious that the characteristics of the detectorsare shifted upward with the increase of 119864
119887119873
0 We show
that for low 119864119887119873
0(0 dB) and for low mutual information
(lt01) MMSE-IC performs better than LC-K-Best decoderHowever with larger mutual information its performance islower Moreover the mutual information of STS-SD is higherthan LC-K-Best decoder and MMSE-IC for different 119864
119887119873
0
For higher 119864119887119873
0(5 dB) MMSE-IC presents lower mutual
information than other decoders when 1198681198601
lt 09Figure 6 shows the EXIT chart for 119864
119887119873
0= 2 dB with
several MIMO detectors namely STS-SD LC-K-Best andMMSE-IC We note that the characteristic of the channeldecoder is independent of 119864
119887119873
0values It is obvious that
the extrinsic mutual information of the channel decoderincreases with the number of iterations We see that after 6 to8 iterations in the case of turbo decoder (Figure 6(a)) thereis no significant improvement on the mutual informationMeanwhile in the case of LDPC decoder in Figure 6(b) 20iterations are enough for LDPC decoder to converge
By comparing the characteristics of STS-SD LC-K-Bestdecoder andMMSE-IC equalizer with both coding schemeswe notice that STS-SD has a larger mutual informationat its output LC-K-Best decoder has slightly less mutualinformation than STS-SD MMSE-IC shows low mutualinformation levels at its output compared to other algorithms
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
STS-SDLC-K-Best
MMSE-IC
EbN0 = 0 2 5dB
5dB
2dB
0dBI E1I
A2
IA1 IE2
Figure 5 Extrinsic information transfer characteristics of MIMOdetectors (STS-SD LC-K-Best andMMSE-IC) at119864
119887119873
0= 0 2 5 dB
in a 4 times 4 MIMO system using 16-QAM
when 1198681198601
lt 09 while for 1198681198601
gt 09 the extrinsic mutualinformation is comparable to others
In the case of turbo decoder (Figure 6(a)) with 119868in =
8 3 outer iterations are sufficient for STS-SD to convergeat 119864
119887119873
0= 2 dB However the same performance can
be attained by performing 4 outer iterations with only 2inner iterations Similarly LC-K-Best decoder shows anequivalent performance but slightly higher119864
119887119873
0is required
The convergence speed of LC-K-Best decoder is a bit lowerthan STS-SD which requires more iterations to get the sameperformance The reason is mainly due to the unreliabilityof LLRs caused by the small list size (L = 16) In thecase of MMSE-IC the characteristic presents a lower mutualinformation than the LC-K-Best decoder when 119868
1198601lt 09
Therefore an equivalent performance can be obtained athigher 119864
119887119873
0or by performing more iterations
In a similar way we study the convergence of MIMOdetection algorithms with LDPC decoder Figure 6(b) showsthe EXIT chart at 119864
119887119873
0= 2 dBThe same conclusion can be
retrieved as in the case of turbo decoder We can see that at119864119887119873
0= 2 dB a clear tunnel is observed between the MIMO
detector and the channel decoder characteristics allowingiterations to bring improvement to the system SimilarlySTS-SD offers higher mutual information than LC-K-Bestdecoder andMMSE-IC equalizer which suggests its superiorsymbol detection performance
Additionally the average decoding trajectory resultingfrom the free-run iterative detection-decoding simulationsis illustrated in Figure 6 at 119864
119887119873
0= 2 dB with 119868out = 4
119868in = 2 and 119868out = 4 119868in = 5 in the case of turbo decoder andLDPC decoder respectively The decoding trajectory closely
10 Mobile Information Systems
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
TurboSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 2 6 8
(a)
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
LDPCSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 5 10 20
(b)
Figure 6 EXIT chart for MIMO detectors (STS-SD LC-K-Best and MMSE-IC) and channel decoder (a) LTE turbo decoder 119870119887
= 1024119877119888
= 12 and (b) LDPC decoder 119873119887
= 1944 119877119888
= 12 at 119864119887119873
0= 2 dB in a 4 times 4 MIMO system using 16-QAM
matches the characteristics in the case of STS-SD and LC-K-Best decoders The little difference from the characteristicsafter a few iterations is due to the correlation of extrinsicinformation caused by the limited interleaver depth In thecase of MMSE-IC the decoding trajectory diverges fromthe characteristics for high mutual information because theequalizer uses the a posteriori information to compute softsymbols instead of the extrinsic information
The best trade-off scheduling of the required numberof iterations is therefore 119868out iterations in the outer loopand a total of 8 iterations inside the turbo decoder and 20iterations inside the LDPC decoder distributed across these119868out iterations
6 Performance and Complexity Evaluation ofIterative Detection-Decoding
In this section we evaluate and compare the performance andthe complexity of differentMIMOdetectors namely STS-SDLC-K-Best decoder MMSE-IC and I-VBLAST equalizerswith different channel coding techniques (turbo LDPC)A detailed analysis of the performance and the complexitytrade-off ofMIMOdetection with LTE turbo decoder and 16-QAM modulation in a Rayleigh channel has been discussedin [24] Herein the performance and the complexity ofthe receiver with LDPC decoder are investigated Moreoverseveral modulations and coding schemes are consideredto quantify the gain achieved by such iterative receiver indifferent channel environments Consequently a comparative
study is conducted in iterative receiver with both codingschemes (turbo LDPC)
For the turbo code the rate 13 turbo encoder specifiedin 3GPP LTE with a block length 119870
119887= 1024 is used in the
simulations Puncturing is performed in the rate matchingmodule to achieve an arbitrary coding rate 119877
119888(eg 119877
119888= 12
34)Meanwhile the LDPC encoder specified in IEEE 80211nis consideredThe encoder is defined by a parity checkmatrixthat is formed out of square submatrices of sizes 27 54 or 81Herein the codeword length of size 119873
119887= 1944 with coding
rate of 119877119888
= 12 and 34 is considered
61 Performance Evaluation The simulations are first carriedout in Rayleigh fading channel to view general performanceof the iterative receivers Real channel models will be consid-ered to evaluate the performance in more realistic scenariosTherefore the 3GPP LTE-(A) channel environments withlow medium and large delay spread values and Dopplerfrequencies are considered The low spread channel is theExtended Pedestrian A (EPA) model which emulates theurban environment with small cell sizes (120591rms = 43 ns) Themedium spread channel (120591rms = 357 ns) is the ExtendedVehicular A (EVA) model The Extended Typical Urban(ETU) model is the large spread channel which has alarger excess delay (120591rms = 991 ns) and simulates extremeurban suburban and rural cases Table 2 summarizes thecharacteristic parameters of these channel environments Forall cases the channel is assumed to be perfectly known atthe receiver Table 3 gives the principle parameters of thesimulations The performance is measured in terms of bit
Mobile Information Systems 11
Table 2 Characteristic parameters of the investigated channelmodels
error rate (BER) with respect to signal-to-noise ratio (SNR)per bit 119864
119887119873
0
119864119887
1198730
=119864119904
1198730
+ 10 sdot log10
1
119877119888119876119873
119905
[dB] (33)
In our previous study [24] the performance of MIMOdetectors with LTE turbo decoder is evaluated in a Rayleighchannel with various outer and inner iterations It has beenshown that the performance is improved by about 15 dB at aBER level of 1 times 10
minus5 with 4 outer iterations It has been alsoshown that no significant improvement can be achieved after4 outer iterations this improvement is less than 02 dB with 8outer iterations
Similarly to turbo decoder we fix the number of inneriterations inside LDPC decoder to 20 while varying thenumber of outer iterations Figure 7 shows BER performanceof MIMO detectors with LDPC decoder in Rayleigh channelwith 119868in = 20 iterations and 119868out = 1 2 4 or 8 iterationsSTS-SD is used with a LLR clipping level of 8 which givesclose toMAPperformancewith considerable reduction in thecomplexity For LC-K-Best decoder a LLR clipping level of 3is used in the case of missing counter hypothesis We show
0 1 2 3 4 5
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
Iout = 4
Iout = 8
Iout = 2
Iout = 1
EbN0 (dB)minus1
Figure 7 BER performance of a 4 times 4 spatial multiplexing systemwith 16-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) in a Rayleigh fading channel 119868out = 1 2 4 8 119868in =
20 LDPC decoder 119873119887
= 1944 and 119877119888
= 12
that performance improvement of 15 dB is observed with 4outer iterations For 119868out = 8 iterations the improvementis less than 02 dB Therefore 119868out = 4 iterations will beconsidered in the sequel
Figure 8 shows the BER performance of 4 times 4 16-QAMsystem in a Rayleigh fading channel with 119868out = 4 and 119868in = 2
in the case of turbo decoder and 119868in = [3 4 6 7] in the caseof LDPC decoder The notation 119868in = [3 4 6 7] denotes that3 inner iterations are performed in the 1st outer iteration 4inner iterations in the 2nd outer iteration and so on Theperformances of STS-SD with 119868in = 8 and 119868in = 20 foreach outer iteration in the case of turbo decoder and LDPCdecoder are also plotted as a reference In the case of turbodecoder we show that performing 119868in = 8 and 119868out = 4
iterations does not bring significant improvement on theperformance compared to the case when 119868in = 2 and 119868out =
4 iterations are performed Similarly in the case of LDPCdecoder the performance of 119868in = 20 and 119868out = 4 iterationsis comparable to the performance of 119868in = [3 4 6 7] and119868out = 4 iterations Hence using a large number of iterationsdoes not seem to be efficientwhich proves the results obtainedin the convergence analysis of Section 5
By comparing the algorithms LC-K-Best decoder showsa degradation of less than 02 dB compared to STS-SD ata BER level of 2 times 10
minus5 However it outperforms MMSE-IC and I-VBLAST equalizer by about 02 dB at a BERlevel of 2 times 10
minus5 MMSE-IC and I-VBLAST show almostthe same performance In addition in the case of LDPCdecoder (Figure 8(b)) we notice that increasing the numberof inner iterations for each outer iteration 119868in = [3 4 6 7]
12 Mobile Information Systems
0 05 1 15 2 25 3
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 8
(a) LTE turbo decoder
BER
0 05 1 15 2 25 3
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 20
STS-SD Iin = 5
STS-SD
LC-K-BestI-VBLASTMMSE-IC
(b) LDPC decoder
Figure 8 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
shows slightly better performance than performing an equalnumber of iterations 119868in = 5 for each outer iteration
In the case of high-order modulation higher spectralefficiency can be achieved at a cost of increased symboldetection difficulty Figure 9 shows the BER performance of64-QAM with 119877
119888= 34 We see that LC-K-Best decoder
with a list size of 32 presents the similar performance as STS-SD However I-VLAST equalizer and MMSE-IC equalizerpresent degradation of more than 2 dB at a BER level of 1 times
10minus5 compared to LC-K-Best decoder Therefore LC-K-Best
decoder is more robust in the case of high-ordermodulationsand high coding rates The figure also shows that the BERperformance of LDPC decoder is almost identical to that ofthe turbo decoder
In order to summarize the performance of differentdetectors with different channel decoders we provide the119864119887119873
0values achieving a BER level of 1 times 10
minus5 in Table 4The values given in the parentheses of the table represent theperformance loss compared to STS-SD
Next we evaluate the performance of the iterative receiverin more realistic channel environments Figures 10 11 and 12show the BER performance of the detectors with the channeldecoders on EPA EVA and ETU channels receptivelySimilar behaviors can be observed with LTE turbo decoderand with LDPC decoder
In EPA channel (Figure 10) we see that LC-K-Bestdecoder achieves similar performance as STS-SD in the caseof 64-AQM and presents a degradation less than 02 dB in the
case of 16-QAM Meanwhile MMSE-IC presents significantperformance loss of more than 6 dB in the case of 64-QAMand 119877
119888= 34 With 16-QAM and 119877
119888= 12 the degradation
of MMSE-IC compared to LC-K-Best decoder is about 1 dBat a BER level of 1 times 10
minus4In EVA channel (Figure 11) the performance loss of
MMSE-IC compared to LC-K-Best decoder is reduced toapproximately 5 dB with 64-QAM and 05 dB with 16-QAMLC-K-Best decoder presents a degradation of about 01sim03 dB compared to STS-SD in the case of 16-QAM and 64-QAM
Similarly in ETU channel (Figure 12) MMSE-IC presentsa performance degradation compared to LC-K-Best decoderThis degradation is less than 4 dB in the case of 64-QAM andless than 05 dB in the case of 16-QAM We notice also thatthe LC-K-Best decoder is comparable to STS-SD in the caseof 64-QAM and has a degradation of 02 dB in the case of 16-QAM
Comparing the performance of the iterative receiver indifferent channels it can be seen that iterative receiverspresent the best performance in ETU channel compared toEPA and EVA channels This is due to the high diversity ofETU channel At a BER level of 1 times 10
minus4 the performancegain in ETU channel in the case of LTE turbo decoder is about08 dB 13 dB compared to EVA channel with 16-QAM and64-QAM respectively In the case of LDPC decoder this gainis 04 dB and 1 dB with 16-QAM and 64-QAM respectivelyHowever in EPA channel the performance gain in ETU
Mobile Information Systems 13
3 4 5 6 7 8 9 10 11 12 13
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
EbN0 (dB)
(a) LTE turbo decoder
3 4 5 6 7 8 9 10 11 12
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)
STS-SD Iin = 20
STS-SDLC-K-Best
I-VBLASTMMSE-IC
(b) LDPC decoder
Figure 9 BER performance of a 4 times 4 spatial multiplexing system with 64-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
0values achieving a BER level of 1times 10minus5 for different detectors and channel decoders (turbo LDPC) in 4times 4 spatialmultiplexing
system with 16-QAM 119877119888
= 12 and 64-QAM 119877119888
= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
STS-SD 25 dB 102 dB 27 dB 96 dBLC-119870-Best 26 dB (minus01) 102 dB (00) 28 dB (minus01) 96 dB (00)I-VBLAST 28 dB (minus03) 128 dB (minus26) 29 dB (minus02) 12 dB (minus24)MMSE-IC 28 dB (minus03) 128 dB (minus26) 30 dB (minus03) 12 dB (minus24)lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
channel in the case of turbo decoder or LDPC decoder ismore than 1 dB with 16-QAM and 64-QAM
Table 5 summarizes the 119864119887119873
0values achieving a BER
level of 1times10minus4 of different detectors combined with different
channel decoders andmodulation orders in different channelmodels The values given in the parentheses in the tablerepresent the performance loss compared to STS-SD Asindicated in Table 5 the iterative receivers with turbo decoderand LDPC decoder have comparable performance with acoding rate 119877
119888= 12 (16-QAM) However with 119877
119888= 34
(64-QAM) the receivers with LDPC decoder present slightlya better performance especially in ETU channel (06 dB)
From these results we show that the iterative receiversubstantially improves the performance of coded MIMOsystems either with turbo decoder or with LDPC decoderin Rayleigh channel (Figures 8 and 9) and in more realisticchannels (Figures 10 11 and 12) Moreover we show that
performing a large number of inner iterations does notbring significant improvement In addition we show thatthe LC-K-Best decoder achieves a good performance withdifferent modulations and channel coding schemes Thefigures suggest that the BER performance of the iterativereceiver with turbo decoder is almost comparable to that ofthe LDPC decoder It is therefore meaningful to evaluate thecomputational complexity of the iterative receivers with bothdecoding techniques as it will be discussed in the next section
62 Complexity Evaluation The computational complexityhas significant impact on the latency throughput and powerconsumption of the deviceTherefore the receiver algorithmsshould be optimized to achieve a good trade-off betweenperformance and cost In this part we evaluate the com-putational complexity of the iterative receiver in terms ofbasic operations such as addition subtractionmultiplication
14 Mobile Information Systems
1 3 5 7 9 11 13 15 17 19
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
(a) EPA turbo 119868out = 4 119868in = 2
BER
1 3 5 7 9 11 13 15 17 19
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12 64-QAM
Rc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EPA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 10 BER performance of a 4 times 4 spatial multiplexing with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) on EPA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944 119868out = 4
119868in = [3 4 6 7]
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
STS-SDLC-K-Best
MMSE-IC
16-QAMRc = 12
64-QAMRc = 34
(a) EVA turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EVA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 11 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Best and MMSE-IC) on EVA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944
and 119868out = 4 119868in = [3 4 6 7]
Mobile Information Systems 15
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(a) ETU turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) ETU LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 12 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SDLC-K-Best and MMSE-IC) on ETU channel model (a) LTE turbo decoder 119870
minus4 in LTE channel models for different detectors and channel decoders (turbo LDPC)in 4 times 4 spatial multiplexing system with 16-QAM 119877
119888= 12 and 64-QAM 119877
119888= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
EPASTS-SD 63 dB 140 dB 62 dB 138 dB
LC-119870-Best 65 dB (minus02) 139 dB (+01) 64 dB (minus02) 139 dB (minus01)MMSE-IC 74 dB (minus11) gt20 dB (gtminus6) 80 dB (minus18) gt20 dB (gtminus6)
EVASTS-SD 52 dB 143 dB 53 dB 134 dB
LC-119870-Best 54 dB (minus02) 144 dB (minus01) 56 dB (minus03) 137 dB (minus03)MMSE-IC 58 dB (minus06) 190 dB (minus47) 60 dB (minus07) 185 dB (minus51)
ETUSTS-SD 44 dB 130 dB 49 dB 124 dB
LC-119870-Best 46 dB (minus02) 130 dB (00) 49 dB (00) 124 dB (00)MMSE-IC 49 dB (minus05) 175 dB (minus35) 51 dB (minus02) 16 dB (minus36)
lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
division square root extraction maximization and look-up table check (which are denoted by ADD SUB MULDIV SQRT Max and LUT resp) To this end the completecomparison of the iterative receivers with both channeldecoders (turbo LDPC) and with several modulations andcoding rates is carried out
621 Complexity of Iterative Receiver The complexity ofiterative receiver depends on theMIMOdetector the channeldecoder and the number of innerouter iterationsThis com-plexity can be expressed by
sdot [119862det1 + (119868out minus 1) sdot 119862deti] (34)
where 119862det1 denotes the complexity of the first iterationof MIMO detection algorithm per symbol vector withouttaking into consideration the a priori information 119862detidenotes the complexity per iteration per symbol vector takinginto consideration the a priori information 119862dec denotesthe complexity of the channel decoder per iteration perinformation bit 119873bit is the number of information bit at theinput of the encoder 119873symb is the number of symbol vectors119873symb and 119873bit are linked by the following relation
119873symb =119873bit
119876119877119888119873119905
= 120572119873bit with 120572 =1
119876119877119888119873119905
(35)
where 119876 is the number of bits in the constellation symbol 119877119888
is the coding rate and119873119905is the number of transmit antennas
16 Mobile Information Systems
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
Figure 3 LDPC decoder (a) matrix representation and Tanner graph and (b) iterative decoding between VNs and CNs
(3) The a posterioriLLRof each codeword bit is computedas
119871119901
(119888119894) = 119871 (119888
119894) + sum
1198951015840isin119862119894
1198711198881198951015840119894
(10)
The decoding algorithm alternates between check node proc-essing and variable node processing until a maximum num-ber of iterations are achieved or until the parity check con-dition is satisfied When the decoding process is terminatedthe decoder outputs the a posteriori LLR
4 Soft-Input Soft-Output MIMO Detection
The aim of MIMO detection is to recover the transmittedvector s from the received vector y The state-of-the-artMIMO detection algorithms have been presented in [24]These algorithms can be divided into two main familiesnamely the tree-search-based detection and the interference-cancellation-based detection In this section we brieflyreview the main existing SISO MIMO detection algorithmsuseful for the following sections
41 MaximumA Posteriori Probability (MAP) Detection TheMAP algorithm achieves the optimum performance throughthe use of an exhaustive search over all 2
119876sdot119873119905 possible symbol
combinations to compute the LLR of each bit The LLR of the119887th bit in the 119894th transmit symbol 119909
119894119887 is given by
119871 (119909119894119887
) = log119875 (119909
119894119887= +1 | y)
119875 (119909119894119887
= minus1 | y)
= logsumsisin120594+1
119894119887
119901 (y | s) 119875 (s)
sumsisin120594minus1119894119887
119901 (y | s) 119875 (s)
(11)
where 120594+1
119894119887and 120594
minus1
119894119887denote the sets of symbol vectors in
which the 119887th bit in the 119894th antenna is equal to +1 and minus1respectively 119901(y | s) is the conditioned probability densityfunction given by
119901 (y | s) =1
(1205871198730)119873119903
exp(minus1
1198730
1003817100381710038171003817y minus Hs10038171003817100381710038172
) (12)
119875(s) represents the a priori information provided by thechannel decoder in the form of a priori LLRs
119871119860
(119909119894119887
) = log119875 (119909
119894119887= +1)
119875 (119909119894119887
= minus1) forall119894 119887
119875 (s) =
119873119905
prod
119894=1
119875 (119904119894) =
119873119905
prod
119894=1
119876
prod
119887=1
119875 (119909119894119887
)
(13)
The max-log-MAP approximation is commonly used in theLLR calculation with lower complexity [12]
119871 (119909119894119887
) asymp1
1198730
min120594minus1
119894119887
1198891 minus
1
1198730
min120594+1
119894119887
1198891 (14)
1198891
=1003817100381710038171003817y minus Hs1003817100381710038171003817
2
minus 1198730
119873119905
sum
119894=1
119876
sum
119887=1
log119875 (119909119894119887
) (15)
where 1198891represents the Euclidean distance between the
received vector y and lattice pointsHsBased on the a posteriori LLRs 119871(119909
119894119887) and the a priori
LLRs 119871119860
(119909119894119887
) the detector computes the extrinsic LLRs119871119864(119909119894119887
) as
119871119864
(119909119894119887
) = 119871 (119909119894119887
) minus 119871119860
(119909119894119887
) (16)
The MAP algorithm is not feasible due to its exponentialcomplexity since 2
119876sdot119873119905 hypotheses have to be considered
within each minimum term and for each bit Thereforeseveral suboptimal MIMO detectors have been proposedwith reduced complexity as will be briefly discussed in thefollowing sections
42 Tree-Search-Based Detection The tree-search-baseddetection methods generally fall into two main categoriesnamely depth-first search like the sphere decoder andbreadth-first search like the K-Best decoder
421 List Sphere Decoder (LSD) The basic idea of the spheredecoder is to limit the search space of the MAP solution to ahypersphere of radius 119903
119904around the received vector Instead
of testing all the hypotheses of the transmitted signal onlythe lattice points that lie inside the hypersphere are testedreducing the computational complexity [33]
sSD = arg minsisin2119876119873119905
1003817100381710038171003817y minus Hs1003817100381710038171003817
2
le 1199032
119904 (17)
6 Mobile Information Systems
Using the QR decomposition in real-valued model thechannel matrix H can be decomposed into two matrixes Qand R (H = QR) where Q is 2119873
119903times 2119873
119905orthogonal matrix
(Q119867Q = I2119873119905
) and R is 2119873119905
times 2119873119905upper triangular matrix
with real-positive diagonal elements [34] Therefore the dis-tance in (17) can be computed as yminusHs2 = yminusRs2 wherey = Q119867y is the modified received symbol vector Exploitingthe triangular nature of R the Euclidean distance metric 119889
1
in (15) can be recursively evaluated through the accumulatedpartial Euclidean distance (PED) 119889
and the a priori-based partial metric at the 119894th level respec-tively
This process can be illustrated by a tree with 2119873119905
+ 1
levels as depicted in Figure 4(a) The tree search starts at theroot level with the first child node at level 2119873
119905 The partial
Euclidean distance 1198892119873119905
in (18) is then computed If 1198892119873119905
issmaller than the sphere radius 119903
119904 the search continues at level
2119873119905minus1 and steps down the tree until finding a valid leaf node
at level 1List sphere decoder is proposed to approximate the MAP
detector [12] It generates a list L sub 2119876119873119905 that includes the
best possible hypotheses The LLR values are then computedfrom this list as
119871 (119909119894119887
) =1
1198730
minLcap120594minus1119894119887
1198891 minus
1
1198730
minLcap120594+1119894119887
1198891 (19)
The main issue of LSD is the missing counter-hypothesisproblem depending on the list sizeThe use of limited list sizecauses inaccurate approximation of the LLR due to missingsome counter hypotheses where no entry can be found inthe list for a particular bit 119909
119894119887= +1 minus1 Several solutions
have been proposed to handle this issue LLR clipping is afrequently used solution which consists simply to set the LLRto a predefined maximum value [12 35]
Several methods can be considered to reduce the com-plexity of the sphere decoder such as Schnorr-Euchner(SE) enumeration [36] layer ordering technique [34] andchannel regularization [37] Layer ordering technique allowsthe selection of the most reliable symbols at a high layerusing the sorted QR (SQR) decomposition However channelregularization introduces a biasing factor in the metricswhich should be removed in LLR computation to avoidperformance degradation as discussed in [38] In the sequelthe SQR decomposition is considered in the preprocessingstep
422 Single Tree-Search SphereDecoder (STS-SD) Oneof thetwominima in (14) corresponds to theMAPhypothesis sMAPwhile the other corresponds to the counter hypothesis Thecomputation of LLR can be expressed by
119871 (119909119894119887
) =
1
1198730
(119889MAP119894119887
minus 119889MAP
) if 119909MAP119894119887
= +1
1
1198730
(119889MAP
minus 119889MAP119894119887
) if 119909MAP119894119887
= minus1
(20)
with
119889MAP
=10038171003817100381710038171003817y minus RsMAP10038171003817100381710038171003817
2
minus 1198730119875 (sMAP
)
119889MAP119894119887
= min119904isin120594
MAP119894119887
1003817100381710038171003817y minus Rs1003817100381710038171003817
2
minus 1198730119875 (s)
sMAP= arg min
119904isin2119876sdot119872
1003817100381710038171003817y minus Rs1003817100381710038171003817
2
minus 1198730119875 (s)
(21)
where 120594MAP119894119887
denotes the bitwise counter hypothesis of theMAP hypothesis which is obtained by searching over all thesolutions with the 119887th bit of the 119894th symbol opposite to thecurrentMAPhypothesisOriginally theMAPhypothesis andthe counter hypotheses can be found through repeating thetree search [39] The repeated tree search yields a large com-putational complexity cost To overcome this the single tree-search algorithm [13 40] was developed to compute all theLLRs concurrently The 119889
MAP metric and the corresponding119889MAP119894119887
metrics are updated through one tree-search processThrough the use of extrinsic LLR clipping method the STS-SD algorithm can be tunable between the MAP performanceand hard-output performance The implementations of STS-SD have been reported in [14 15]
423 SISO K-Best Decoder K-Best algorithm is a breadth-first search based algorithm in which the tree is traversedonly in the forward direction [41] This approach searchesonly a fixed number 119870 of paths with best metrics at eachdetection layer Figure 4(b) shows an example of the treesearch with 119873
119905= 2 The algorithm starts by extending the
root node to all possible candidates It then sorts the newpaths according to their metrics and retains the 119870 paths withsmallest metrics for the next detection layer
K-Best algorithm is able to achieve near-optimal perfor-mance with a fixed and affordable complexity for parallelimplementation Yet the major drawbacks ofK-Best decoderare the expansion and the sorting operations that are verytime consuming Several proposals have been drawn in theliterature to approximate the sorting operations such asrelaxed sorting [42] and distributed sorting [43] or evento avoid sorting using on-demand expansion scheme [44]Moreover similarly as LSD K-Best decoder suffers frommissing counter-hypothesis problem due to the limited listsize Numerous approaches have been proposed to addressthis problem such as smart candidates adding [45] bit flip-ping [46] and path augmentation and LLR clipping [12 35]
43 Interference-Cancellation-Based Detection Interference-cancellation-based detection can be carried out either in aparallel way as inMMSE-IC [8 9] or in a successive way as inVBLAST [47]
431 Minimum Mean Square Error-Interference Cancella-tion (MMSE-IC) Equalizer MMSE-IC equalizer can be per-formed using two filters [4]The first filter p
119894is applied to the
received vector y and the second filter q119894is applied to the
estimated vector s in order to cancel the interference fromother layers The equalized symbol
119894can be written as
119894= p119867
119894y minus q119867
119894s119894
with 119894 isin [1 119873119905] (22)
where s119894denotes the estimated vector given by the
previous iteration with the 119894th symbol omitteds119894
= [1 sdot sdot sdot 119894minus1
0 119894+1
sdot sdot sdot 119873119905]
119894is calculated by the
soft mapper as 119894
= E[119904119894] = sum
119904isin2119876 119904119875 (119904
119894= 119904) [48] The filters
p119894and q
119894are optimized using the MMSE criterion and are
given in [6 24]For the first iteration since no a priori information is
available the equalization process is reduced to the classicalMMSE solution
119894= (H119867H +
1205902
119899
1205902119904
I119873119905
)
minus1
H119867y (23)
The equalized symbols 119894are associated with a bias factor 120573
119894
in addition to some residual noise plus interferences 120578119894
119894= 120573
119894119904119894+ 120578
119894 (24)
These equalized symbols are then used by the soft demapperto compute the LLR values using the max-log-MAP approxi-mation [48]
119871 (119909119894119887
) =1
1205902120578119894
(min119904119894isin120594minus1
119894119887
1003816100381610038161003816119894 minus 120573119894119904119894
1003816100381610038161003816
2
minus min119904119894isin120594+1
119894119887
1003816100381610038161003816119894 minus 120573119894119904119894
1003816100381610038161003816
2
) (25)
MMSE-IC equalizer requires 119873119905matrix inversions for each
symbol vector For this reason several approximations of
MMSE-IC were proposed For example in [9] a low-complexity approach of MMSE-IC is described by perform-ing a single matrix inversion without performance loss Thisalgorithm is referred to as LC-MMSE-IC
432 Successive Interference Cancellation (SIC) EqualizerThe SIC-based detector was initially used in the VBLASTsystems In VBLAST architecture [47] a successive cancel-lation step followed by an interference nulling step is usedto detect the transmitted symbols However this methodsuffers from error propagation An improved VBLAST foriterative detection and decoding is described in [49] At thefirst iteration an enhanced VBLAST which takes decisionerrors into account is employed [24] When the a prioriLLRs are available from the channel decoder soft symbols arecomputed by a soft mapper and are used in the interferencecancellation To describe the enhanced VBLAST algorithmwe assume that the detection order has been made accordingto the optimal detection order [47] For the 119894th step thepredetected symbol vector s
119894minus1until step 119894 minus 1 is canceled out
from the received signal
y119894= y minus H
1119894minus1s119894minus1
(26)
where s119894minus1
= [1
2
sdot sdot sdot 119894minus1
] and H1119894minus1
=
[h1h2
sdot sdot sdot h119894minus1
] with h119894being the 119894th column of H
Then the estimated symbol 119894is obtained using a filtered
matrix W119894based on the MMSE criterion that takes decision
errors into account [11 49]
119894= W119867
119894y119894= 120573
119894119904119894+ 120578
119894
W119894= 120590
2
119904(HΣ
119894H119867 + 119873
0I119873119903
)minus1
h119894
(27)
Σ119894is the decision error covariance matrix defined as
Σ119894=
119894minus1
sum
119895=1
1205982
119895e119895e119879119895
+
119873119905minus119894+1
sum
119895=119894
1205902
119904e119895e119879119895
1205982
119895= E
10038161003816100381610038161003816119904119895
minus 119895
10038161003816100381610038161003816
2
| 119895minus1
(28)
8 Mobile Information Systems
where e119894denotes a unit vector having zero components except
the 119894th component which is oneA soft demapper is then used to compute LLRs according
to (25) We refer to this algorithm as improved VBLAST (I-VBLAST) in the sequel
44 Low-Complexity K-Best Decoder The low-complexityK-Best (LC-K-Best) decoder recently proposed in [20] usestwo improvements over the classical K-Best decoder for thesake of lower complexity and latency The first improvementsimplifies the hybrid enumeration of the constellation pointsin real-valued systemmodel when the a priori information isincorporated into the tree search using two look-up tablesThe second improvement is to use a relaxed on-demandexpansion that reduces the need of exhaustive expansionand sorting operations The LC-K-Best algorithm can bedescribed as follows
The preprocessing step is as follows
(1) Input yH 119870CalculateH = QR y = Q119867y
(2) Enumerate the constellation symbols based on119898119860 for
all layers
The tree-search step is as follows
(1) Set layer to 2119873119905 listL = 0 119889
2119873119905+1
= 0
(a) expand all radic2119876 possible constellation nodes(b) calculate the corresponding PEDs(c) if radic2119876 gt 119870 select the 119870 best nodes and store
them in the listL2119873119905
(2) For layer 119894 = 2119873119905minus 1 1
(a) enumerate the constellation point according to119898119862
119894of the 119870 surviving paths in the listL
119894+1
(b) find the first child (FC) based on 119898119862
119894and 119898
119860
119894for
each parent nodes(c) compute their PEDs(d) select 119860 best children with smallest PEDs
among the 119870 FCs and add them to the listL119894
(e) if |L119894| lt 119870 find the next child (NC) of the
selected parent nodesCalculate their PEDs and go to step (2)(d)
(f) else move to the next layer 119894 = 119894 minus 1 and go tostep (2)
(3) If 119894 = 1 calculate the LLR as in (19)In the case of missing counter hypothesis LLR clip-ping method is used
It has been shown in [20] that the LC-K-Best decoderachieves almost the same performance as the classical K-Best decoder with different modulations Moreover thecomputational complexity in terms of the number of visitednodes is significantly reduced specially in the case of high-order modulations
5 Convergence of IterativeDetection-Decoding
The EXtrinsic Information Transfer (EXIT) chart is a usefultool to study the convergence behavior of iterative decod-ing systems [50] It describes the exchange of the mutualinformation in the iterative process in order to predict therequired number of iterations the convergence threshold(corresponding to the start of the waterfall region) and theaverage decoding trajectory
In the iterative receiver considered in our study two itera-tive processes are performed one inside the channel decoder(turbo or LDPC) and the other between the MIMO detectorand the channel decoder For simplicity we separately studythe convergence of the channel decoding and the MIMOdetection We denote by 119868
1198601and 119868
1198602the a priorimutual input
information of the MIMO detector and the channel decoderrespectively and by 119868
1198641and 119868
1198642their corresponding extrinsic
mutual output informationThe mutual information 119868
119909(119868119860or 119868
119864) can be computed
through Monte Carlo simulation using the probability den-sity function 119901
119871119909
[50]
119868119909
=1
2sum
119909isinminus11
int
+infin
minusinfin
119901119871119909
(119871119909
| 119909)
sdot log2
2119901119871119909
(119871119909
| 119909)
119901119871119909
(119871119909
| minus1) 119901119871119909
(119871119909
| +1)119889119871
119909
(29)
A simple approximation of the mutual information is used inour analysis [51]
119868119909
asymp 1 minus1
119871119887
119871119887minus1
sum
119899=0
log2
(1 + exp (minus119909119871119909)) (30)
where 119871119887is the number of transmitted bits and 119871
119909is the LLR
associated with the bit 119909 isin minus1 +1The a priori information 119871
119860can be modeled by applying
an independentGaussian randomvariable 119899119860with zeromean
and variance 1205902
119860in conjunction with the known transmitted
information bits 119909 [50]
119871119860
= 120583119860
119909 + 119899119860
where 120583119860
=1205902
119860
2 (31)
For each given mutual information value 119868119860
isin [0 1] 1205902
119860can
be computed using the following equation [52]
120590119860
asymp (minus1
1198671
log2
(1 minus 11986811198673
119860))
121198672
(32)
where 1198671
= 03073 1198672
= 08935 and 1198673
= 11064At the beginning the a priori mutual information is as
follows 1198681198601
= 0 and 1198681198602
= 0 Then the extrinsic mutualinformation 119868
1198641of the MIMO detector becomes the a priori
mutual information 1198681198602
of the channel decoder and so onand so forth (ie 119868
1198641= 119868
1198602and 119868
1198642= 119868
1198601) For a successful
decoding there must be an open tunnel between the curves
Mobile Information Systems 9
Table 1 Simulation parameters for convergence analysis
MIMO system 4 times 4 spatial multiplexingModulation 16-QAM Gray mappingChannel Flat Rayleigh fading
DetectorSTS-SD 119871 clip = plusmn8
LC-119870-Best 119870 = 16 119871 clip = plusmn3
MMSE-IC
Channel decoder
LTE turbo code [13 15]119900
Block length 119870119887
= 1024 119877119888
= 12
LDPC code (IEEE 80211n)Codeword length 119873
119887= 1944 119877
119888= 12
Interleaver Random size = 1024 (turbo code case)
the exchange of extrinsic information can be visualized as aldquozigzagrdquo decoding trajectory in the EXIT chart
To visualize the exchange of extrinsic information of theiterative receiver we present the MIMO detector and thechannel decoder characteristics into a single chart For ourconvergence analysis a 4 times 4 MIMO system with 16-QAMconstellation turbo decoder and LDPCdecoder (119877
119888= 12) is
considered Table 1 summarizes the main parameters for theconvergence analysis
Figure 5 shows the extrinsic information transfer char-acteristics of MIMO detectors at different 119864
119887119873
0values
As the I-VBLAST detector performs successive interferencecancellation at the first iteration and parallel interference can-cellation of the soft estimated symbols for the rest iterationsit is less intuitive to present its convergence in the EXITchart Therefore the convergence analysis of VBLAST is notconsidered
It is obvious that the characteristics of the detectorsare shifted upward with the increase of 119864
119887119873
0 We show
that for low 119864119887119873
0(0 dB) and for low mutual information
(lt01) MMSE-IC performs better than LC-K-Best decoderHowever with larger mutual information its performance islower Moreover the mutual information of STS-SD is higherthan LC-K-Best decoder and MMSE-IC for different 119864
119887119873
0
For higher 119864119887119873
0(5 dB) MMSE-IC presents lower mutual
information than other decoders when 1198681198601
lt 09Figure 6 shows the EXIT chart for 119864
119887119873
0= 2 dB with
several MIMO detectors namely STS-SD LC-K-Best andMMSE-IC We note that the characteristic of the channeldecoder is independent of 119864
119887119873
0values It is obvious that
the extrinsic mutual information of the channel decoderincreases with the number of iterations We see that after 6 to8 iterations in the case of turbo decoder (Figure 6(a)) thereis no significant improvement on the mutual informationMeanwhile in the case of LDPC decoder in Figure 6(b) 20iterations are enough for LDPC decoder to converge
By comparing the characteristics of STS-SD LC-K-Bestdecoder andMMSE-IC equalizer with both coding schemeswe notice that STS-SD has a larger mutual informationat its output LC-K-Best decoder has slightly less mutualinformation than STS-SD MMSE-IC shows low mutualinformation levels at its output compared to other algorithms
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
STS-SDLC-K-Best
MMSE-IC
EbN0 = 0 2 5dB
5dB
2dB
0dBI E1I
A2
IA1 IE2
Figure 5 Extrinsic information transfer characteristics of MIMOdetectors (STS-SD LC-K-Best andMMSE-IC) at119864
119887119873
0= 0 2 5 dB
in a 4 times 4 MIMO system using 16-QAM
when 1198681198601
lt 09 while for 1198681198601
gt 09 the extrinsic mutualinformation is comparable to others
In the case of turbo decoder (Figure 6(a)) with 119868in =
8 3 outer iterations are sufficient for STS-SD to convergeat 119864
119887119873
0= 2 dB However the same performance can
be attained by performing 4 outer iterations with only 2inner iterations Similarly LC-K-Best decoder shows anequivalent performance but slightly higher119864
119887119873
0is required
The convergence speed of LC-K-Best decoder is a bit lowerthan STS-SD which requires more iterations to get the sameperformance The reason is mainly due to the unreliabilityof LLRs caused by the small list size (L = 16) In thecase of MMSE-IC the characteristic presents a lower mutualinformation than the LC-K-Best decoder when 119868
1198601lt 09
Therefore an equivalent performance can be obtained athigher 119864
119887119873
0or by performing more iterations
In a similar way we study the convergence of MIMOdetection algorithms with LDPC decoder Figure 6(b) showsthe EXIT chart at 119864
119887119873
0= 2 dBThe same conclusion can be
retrieved as in the case of turbo decoder We can see that at119864119887119873
0= 2 dB a clear tunnel is observed between the MIMO
detector and the channel decoder characteristics allowingiterations to bring improvement to the system SimilarlySTS-SD offers higher mutual information than LC-K-Bestdecoder andMMSE-IC equalizer which suggests its superiorsymbol detection performance
Additionally the average decoding trajectory resultingfrom the free-run iterative detection-decoding simulationsis illustrated in Figure 6 at 119864
119887119873
0= 2 dB with 119868out = 4
119868in = 2 and 119868out = 4 119868in = 5 in the case of turbo decoder andLDPC decoder respectively The decoding trajectory closely
10 Mobile Information Systems
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
TurboSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 2 6 8
(a)
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
LDPCSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 5 10 20
(b)
Figure 6 EXIT chart for MIMO detectors (STS-SD LC-K-Best and MMSE-IC) and channel decoder (a) LTE turbo decoder 119870119887
= 1024119877119888
= 12 and (b) LDPC decoder 119873119887
= 1944 119877119888
= 12 at 119864119887119873
0= 2 dB in a 4 times 4 MIMO system using 16-QAM
matches the characteristics in the case of STS-SD and LC-K-Best decoders The little difference from the characteristicsafter a few iterations is due to the correlation of extrinsicinformation caused by the limited interleaver depth In thecase of MMSE-IC the decoding trajectory diverges fromthe characteristics for high mutual information because theequalizer uses the a posteriori information to compute softsymbols instead of the extrinsic information
The best trade-off scheduling of the required numberof iterations is therefore 119868out iterations in the outer loopand a total of 8 iterations inside the turbo decoder and 20iterations inside the LDPC decoder distributed across these119868out iterations
6 Performance and Complexity Evaluation ofIterative Detection-Decoding
In this section we evaluate and compare the performance andthe complexity of differentMIMOdetectors namely STS-SDLC-K-Best decoder MMSE-IC and I-VBLAST equalizerswith different channel coding techniques (turbo LDPC)A detailed analysis of the performance and the complexitytrade-off ofMIMOdetection with LTE turbo decoder and 16-QAM modulation in a Rayleigh channel has been discussedin [24] Herein the performance and the complexity ofthe receiver with LDPC decoder are investigated Moreoverseveral modulations and coding schemes are consideredto quantify the gain achieved by such iterative receiver indifferent channel environments Consequently a comparative
study is conducted in iterative receiver with both codingschemes (turbo LDPC)
For the turbo code the rate 13 turbo encoder specifiedin 3GPP LTE with a block length 119870
119887= 1024 is used in the
simulations Puncturing is performed in the rate matchingmodule to achieve an arbitrary coding rate 119877
119888(eg 119877
119888= 12
34)Meanwhile the LDPC encoder specified in IEEE 80211nis consideredThe encoder is defined by a parity checkmatrixthat is formed out of square submatrices of sizes 27 54 or 81Herein the codeword length of size 119873
119887= 1944 with coding
rate of 119877119888
= 12 and 34 is considered
61 Performance Evaluation The simulations are first carriedout in Rayleigh fading channel to view general performanceof the iterative receivers Real channel models will be consid-ered to evaluate the performance in more realistic scenariosTherefore the 3GPP LTE-(A) channel environments withlow medium and large delay spread values and Dopplerfrequencies are considered The low spread channel is theExtended Pedestrian A (EPA) model which emulates theurban environment with small cell sizes (120591rms = 43 ns) Themedium spread channel (120591rms = 357 ns) is the ExtendedVehicular A (EVA) model The Extended Typical Urban(ETU) model is the large spread channel which has alarger excess delay (120591rms = 991 ns) and simulates extremeurban suburban and rural cases Table 2 summarizes thecharacteristic parameters of these channel environments Forall cases the channel is assumed to be perfectly known atthe receiver Table 3 gives the principle parameters of thesimulations The performance is measured in terms of bit
Mobile Information Systems 11
Table 2 Characteristic parameters of the investigated channelmodels
error rate (BER) with respect to signal-to-noise ratio (SNR)per bit 119864
119887119873
0
119864119887
1198730
=119864119904
1198730
+ 10 sdot log10
1
119877119888119876119873
119905
[dB] (33)
In our previous study [24] the performance of MIMOdetectors with LTE turbo decoder is evaluated in a Rayleighchannel with various outer and inner iterations It has beenshown that the performance is improved by about 15 dB at aBER level of 1 times 10
minus5 with 4 outer iterations It has been alsoshown that no significant improvement can be achieved after4 outer iterations this improvement is less than 02 dB with 8outer iterations
Similarly to turbo decoder we fix the number of inneriterations inside LDPC decoder to 20 while varying thenumber of outer iterations Figure 7 shows BER performanceof MIMO detectors with LDPC decoder in Rayleigh channelwith 119868in = 20 iterations and 119868out = 1 2 4 or 8 iterationsSTS-SD is used with a LLR clipping level of 8 which givesclose toMAPperformancewith considerable reduction in thecomplexity For LC-K-Best decoder a LLR clipping level of 3is used in the case of missing counter hypothesis We show
0 1 2 3 4 5
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
Iout = 4
Iout = 8
Iout = 2
Iout = 1
EbN0 (dB)minus1
Figure 7 BER performance of a 4 times 4 spatial multiplexing systemwith 16-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) in a Rayleigh fading channel 119868out = 1 2 4 8 119868in =
20 LDPC decoder 119873119887
= 1944 and 119877119888
= 12
that performance improvement of 15 dB is observed with 4outer iterations For 119868out = 8 iterations the improvementis less than 02 dB Therefore 119868out = 4 iterations will beconsidered in the sequel
Figure 8 shows the BER performance of 4 times 4 16-QAMsystem in a Rayleigh fading channel with 119868out = 4 and 119868in = 2
in the case of turbo decoder and 119868in = [3 4 6 7] in the caseof LDPC decoder The notation 119868in = [3 4 6 7] denotes that3 inner iterations are performed in the 1st outer iteration 4inner iterations in the 2nd outer iteration and so on Theperformances of STS-SD with 119868in = 8 and 119868in = 20 foreach outer iteration in the case of turbo decoder and LDPCdecoder are also plotted as a reference In the case of turbodecoder we show that performing 119868in = 8 and 119868out = 4
iterations does not bring significant improvement on theperformance compared to the case when 119868in = 2 and 119868out =
4 iterations are performed Similarly in the case of LDPCdecoder the performance of 119868in = 20 and 119868out = 4 iterationsis comparable to the performance of 119868in = [3 4 6 7] and119868out = 4 iterations Hence using a large number of iterationsdoes not seem to be efficientwhich proves the results obtainedin the convergence analysis of Section 5
By comparing the algorithms LC-K-Best decoder showsa degradation of less than 02 dB compared to STS-SD ata BER level of 2 times 10
minus5 However it outperforms MMSE-IC and I-VBLAST equalizer by about 02 dB at a BERlevel of 2 times 10
minus5 MMSE-IC and I-VBLAST show almostthe same performance In addition in the case of LDPCdecoder (Figure 8(b)) we notice that increasing the numberof inner iterations for each outer iteration 119868in = [3 4 6 7]
12 Mobile Information Systems
0 05 1 15 2 25 3
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 8
(a) LTE turbo decoder
BER
0 05 1 15 2 25 3
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 20
STS-SD Iin = 5
STS-SD
LC-K-BestI-VBLASTMMSE-IC
(b) LDPC decoder
Figure 8 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
shows slightly better performance than performing an equalnumber of iterations 119868in = 5 for each outer iteration
In the case of high-order modulation higher spectralefficiency can be achieved at a cost of increased symboldetection difficulty Figure 9 shows the BER performance of64-QAM with 119877
119888= 34 We see that LC-K-Best decoder
with a list size of 32 presents the similar performance as STS-SD However I-VLAST equalizer and MMSE-IC equalizerpresent degradation of more than 2 dB at a BER level of 1 times
10minus5 compared to LC-K-Best decoder Therefore LC-K-Best
decoder is more robust in the case of high-ordermodulationsand high coding rates The figure also shows that the BERperformance of LDPC decoder is almost identical to that ofthe turbo decoder
In order to summarize the performance of differentdetectors with different channel decoders we provide the119864119887119873
0values achieving a BER level of 1 times 10
minus5 in Table 4The values given in the parentheses of the table represent theperformance loss compared to STS-SD
Next we evaluate the performance of the iterative receiverin more realistic channel environments Figures 10 11 and 12show the BER performance of the detectors with the channeldecoders on EPA EVA and ETU channels receptivelySimilar behaviors can be observed with LTE turbo decoderand with LDPC decoder
In EPA channel (Figure 10) we see that LC-K-Bestdecoder achieves similar performance as STS-SD in the caseof 64-AQM and presents a degradation less than 02 dB in the
case of 16-QAM Meanwhile MMSE-IC presents significantperformance loss of more than 6 dB in the case of 64-QAMand 119877
119888= 34 With 16-QAM and 119877
119888= 12 the degradation
of MMSE-IC compared to LC-K-Best decoder is about 1 dBat a BER level of 1 times 10
minus4In EVA channel (Figure 11) the performance loss of
MMSE-IC compared to LC-K-Best decoder is reduced toapproximately 5 dB with 64-QAM and 05 dB with 16-QAMLC-K-Best decoder presents a degradation of about 01sim03 dB compared to STS-SD in the case of 16-QAM and 64-QAM
Similarly in ETU channel (Figure 12) MMSE-IC presentsa performance degradation compared to LC-K-Best decoderThis degradation is less than 4 dB in the case of 64-QAM andless than 05 dB in the case of 16-QAM We notice also thatthe LC-K-Best decoder is comparable to STS-SD in the caseof 64-QAM and has a degradation of 02 dB in the case of 16-QAM
Comparing the performance of the iterative receiver indifferent channels it can be seen that iterative receiverspresent the best performance in ETU channel compared toEPA and EVA channels This is due to the high diversity ofETU channel At a BER level of 1 times 10
minus4 the performancegain in ETU channel in the case of LTE turbo decoder is about08 dB 13 dB compared to EVA channel with 16-QAM and64-QAM respectively In the case of LDPC decoder this gainis 04 dB and 1 dB with 16-QAM and 64-QAM respectivelyHowever in EPA channel the performance gain in ETU
Mobile Information Systems 13
3 4 5 6 7 8 9 10 11 12 13
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
EbN0 (dB)
(a) LTE turbo decoder
3 4 5 6 7 8 9 10 11 12
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)
STS-SD Iin = 20
STS-SDLC-K-Best
I-VBLASTMMSE-IC
(b) LDPC decoder
Figure 9 BER performance of a 4 times 4 spatial multiplexing system with 64-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
0values achieving a BER level of 1times 10minus5 for different detectors and channel decoders (turbo LDPC) in 4times 4 spatialmultiplexing
system with 16-QAM 119877119888
= 12 and 64-QAM 119877119888
= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
STS-SD 25 dB 102 dB 27 dB 96 dBLC-119870-Best 26 dB (minus01) 102 dB (00) 28 dB (minus01) 96 dB (00)I-VBLAST 28 dB (minus03) 128 dB (minus26) 29 dB (minus02) 12 dB (minus24)MMSE-IC 28 dB (minus03) 128 dB (minus26) 30 dB (minus03) 12 dB (minus24)lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
channel in the case of turbo decoder or LDPC decoder ismore than 1 dB with 16-QAM and 64-QAM
Table 5 summarizes the 119864119887119873
0values achieving a BER
level of 1times10minus4 of different detectors combined with different
channel decoders andmodulation orders in different channelmodels The values given in the parentheses in the tablerepresent the performance loss compared to STS-SD Asindicated in Table 5 the iterative receivers with turbo decoderand LDPC decoder have comparable performance with acoding rate 119877
119888= 12 (16-QAM) However with 119877
119888= 34
(64-QAM) the receivers with LDPC decoder present slightlya better performance especially in ETU channel (06 dB)
From these results we show that the iterative receiversubstantially improves the performance of coded MIMOsystems either with turbo decoder or with LDPC decoderin Rayleigh channel (Figures 8 and 9) and in more realisticchannels (Figures 10 11 and 12) Moreover we show that
performing a large number of inner iterations does notbring significant improvement In addition we show thatthe LC-K-Best decoder achieves a good performance withdifferent modulations and channel coding schemes Thefigures suggest that the BER performance of the iterativereceiver with turbo decoder is almost comparable to that ofthe LDPC decoder It is therefore meaningful to evaluate thecomputational complexity of the iterative receivers with bothdecoding techniques as it will be discussed in the next section
62 Complexity Evaluation The computational complexityhas significant impact on the latency throughput and powerconsumption of the deviceTherefore the receiver algorithmsshould be optimized to achieve a good trade-off betweenperformance and cost In this part we evaluate the com-putational complexity of the iterative receiver in terms ofbasic operations such as addition subtractionmultiplication
14 Mobile Information Systems
1 3 5 7 9 11 13 15 17 19
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
(a) EPA turbo 119868out = 4 119868in = 2
BER
1 3 5 7 9 11 13 15 17 19
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12 64-QAM
Rc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EPA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 10 BER performance of a 4 times 4 spatial multiplexing with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) on EPA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944 119868out = 4
119868in = [3 4 6 7]
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
STS-SDLC-K-Best
MMSE-IC
16-QAMRc = 12
64-QAMRc = 34
(a) EVA turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EVA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 11 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Best and MMSE-IC) on EVA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944
and 119868out = 4 119868in = [3 4 6 7]
Mobile Information Systems 15
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(a) ETU turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) ETU LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 12 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SDLC-K-Best and MMSE-IC) on ETU channel model (a) LTE turbo decoder 119870
minus4 in LTE channel models for different detectors and channel decoders (turbo LDPC)in 4 times 4 spatial multiplexing system with 16-QAM 119877
119888= 12 and 64-QAM 119877
119888= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
EPASTS-SD 63 dB 140 dB 62 dB 138 dB
LC-119870-Best 65 dB (minus02) 139 dB (+01) 64 dB (minus02) 139 dB (minus01)MMSE-IC 74 dB (minus11) gt20 dB (gtminus6) 80 dB (minus18) gt20 dB (gtminus6)
EVASTS-SD 52 dB 143 dB 53 dB 134 dB
LC-119870-Best 54 dB (minus02) 144 dB (minus01) 56 dB (minus03) 137 dB (minus03)MMSE-IC 58 dB (minus06) 190 dB (minus47) 60 dB (minus07) 185 dB (minus51)
ETUSTS-SD 44 dB 130 dB 49 dB 124 dB
LC-119870-Best 46 dB (minus02) 130 dB (00) 49 dB (00) 124 dB (00)MMSE-IC 49 dB (minus05) 175 dB (minus35) 51 dB (minus02) 16 dB (minus36)
lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
division square root extraction maximization and look-up table check (which are denoted by ADD SUB MULDIV SQRT Max and LUT resp) To this end the completecomparison of the iterative receivers with both channeldecoders (turbo LDPC) and with several modulations andcoding rates is carried out
621 Complexity of Iterative Receiver The complexity ofiterative receiver depends on theMIMOdetector the channeldecoder and the number of innerouter iterationsThis com-plexity can be expressed by
sdot [119862det1 + (119868out minus 1) sdot 119862deti] (34)
where 119862det1 denotes the complexity of the first iterationof MIMO detection algorithm per symbol vector withouttaking into consideration the a priori information 119862detidenotes the complexity per iteration per symbol vector takinginto consideration the a priori information 119862dec denotesthe complexity of the channel decoder per iteration perinformation bit 119873bit is the number of information bit at theinput of the encoder 119873symb is the number of symbol vectors119873symb and 119873bit are linked by the following relation
119873symb =119873bit
119876119877119888119873119905
= 120572119873bit with 120572 =1
119876119877119888119873119905
(35)
where 119876 is the number of bits in the constellation symbol 119877119888
is the coding rate and119873119905is the number of transmit antennas
16 Mobile Information Systems
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
Using the QR decomposition in real-valued model thechannel matrix H can be decomposed into two matrixes Qand R (H = QR) where Q is 2119873
119903times 2119873
119905orthogonal matrix
(Q119867Q = I2119873119905
) and R is 2119873119905
times 2119873119905upper triangular matrix
with real-positive diagonal elements [34] Therefore the dis-tance in (17) can be computed as yminusHs2 = yminusRs2 wherey = Q119867y is the modified received symbol vector Exploitingthe triangular nature of R the Euclidean distance metric 119889
1
in (15) can be recursively evaluated through the accumulatedpartial Euclidean distance (PED) 119889
and the a priori-based partial metric at the 119894th level respec-tively
This process can be illustrated by a tree with 2119873119905
+ 1
levels as depicted in Figure 4(a) The tree search starts at theroot level with the first child node at level 2119873
119905 The partial
Euclidean distance 1198892119873119905
in (18) is then computed If 1198892119873119905
issmaller than the sphere radius 119903
119904 the search continues at level
2119873119905minus1 and steps down the tree until finding a valid leaf node
at level 1List sphere decoder is proposed to approximate the MAP
detector [12] It generates a list L sub 2119876119873119905 that includes the
best possible hypotheses The LLR values are then computedfrom this list as
119871 (119909119894119887
) =1
1198730
minLcap120594minus1119894119887
1198891 minus
1
1198730
minLcap120594+1119894119887
1198891 (19)
The main issue of LSD is the missing counter-hypothesisproblem depending on the list sizeThe use of limited list sizecauses inaccurate approximation of the LLR due to missingsome counter hypotheses where no entry can be found inthe list for a particular bit 119909
119894119887= +1 minus1 Several solutions
have been proposed to handle this issue LLR clipping is afrequently used solution which consists simply to set the LLRto a predefined maximum value [12 35]
Several methods can be considered to reduce the com-plexity of the sphere decoder such as Schnorr-Euchner(SE) enumeration [36] layer ordering technique [34] andchannel regularization [37] Layer ordering technique allowsthe selection of the most reliable symbols at a high layerusing the sorted QR (SQR) decomposition However channelregularization introduces a biasing factor in the metricswhich should be removed in LLR computation to avoidperformance degradation as discussed in [38] In the sequelthe SQR decomposition is considered in the preprocessingstep
422 Single Tree-Search SphereDecoder (STS-SD) Oneof thetwominima in (14) corresponds to theMAPhypothesis sMAPwhile the other corresponds to the counter hypothesis Thecomputation of LLR can be expressed by
119871 (119909119894119887
) =
1
1198730
(119889MAP119894119887
minus 119889MAP
) if 119909MAP119894119887
= +1
1
1198730
(119889MAP
minus 119889MAP119894119887
) if 119909MAP119894119887
= minus1
(20)
with
119889MAP
=10038171003817100381710038171003817y minus RsMAP10038171003817100381710038171003817
2
minus 1198730119875 (sMAP
)
119889MAP119894119887
= min119904isin120594
MAP119894119887
1003817100381710038171003817y minus Rs1003817100381710038171003817
2
minus 1198730119875 (s)
sMAP= arg min
119904isin2119876sdot119872
1003817100381710038171003817y minus Rs1003817100381710038171003817
2
minus 1198730119875 (s)
(21)
where 120594MAP119894119887
denotes the bitwise counter hypothesis of theMAP hypothesis which is obtained by searching over all thesolutions with the 119887th bit of the 119894th symbol opposite to thecurrentMAPhypothesisOriginally theMAPhypothesis andthe counter hypotheses can be found through repeating thetree search [39] The repeated tree search yields a large com-putational complexity cost To overcome this the single tree-search algorithm [13 40] was developed to compute all theLLRs concurrently The 119889
MAP metric and the corresponding119889MAP119894119887
metrics are updated through one tree-search processThrough the use of extrinsic LLR clipping method the STS-SD algorithm can be tunable between the MAP performanceand hard-output performance The implementations of STS-SD have been reported in [14 15]
423 SISO K-Best Decoder K-Best algorithm is a breadth-first search based algorithm in which the tree is traversedonly in the forward direction [41] This approach searchesonly a fixed number 119870 of paths with best metrics at eachdetection layer Figure 4(b) shows an example of the treesearch with 119873
119905= 2 The algorithm starts by extending the
root node to all possible candidates It then sorts the newpaths according to their metrics and retains the 119870 paths withsmallest metrics for the next detection layer
K-Best algorithm is able to achieve near-optimal perfor-mance with a fixed and affordable complexity for parallelimplementation Yet the major drawbacks ofK-Best decoderare the expansion and the sorting operations that are verytime consuming Several proposals have been drawn in theliterature to approximate the sorting operations such asrelaxed sorting [42] and distributed sorting [43] or evento avoid sorting using on-demand expansion scheme [44]Moreover similarly as LSD K-Best decoder suffers frommissing counter-hypothesis problem due to the limited listsize Numerous approaches have been proposed to addressthis problem such as smart candidates adding [45] bit flip-ping [46] and path augmentation and LLR clipping [12 35]
43 Interference-Cancellation-Based Detection Interference-cancellation-based detection can be carried out either in aparallel way as inMMSE-IC [8 9] or in a successive way as inVBLAST [47]
431 Minimum Mean Square Error-Interference Cancella-tion (MMSE-IC) Equalizer MMSE-IC equalizer can be per-formed using two filters [4]The first filter p
119894is applied to the
received vector y and the second filter q119894is applied to the
estimated vector s in order to cancel the interference fromother layers The equalized symbol
119894can be written as
119894= p119867
119894y minus q119867
119894s119894
with 119894 isin [1 119873119905] (22)
where s119894denotes the estimated vector given by the
previous iteration with the 119894th symbol omitteds119894
= [1 sdot sdot sdot 119894minus1
0 119894+1
sdot sdot sdot 119873119905]
119894is calculated by the
soft mapper as 119894
= E[119904119894] = sum
119904isin2119876 119904119875 (119904
119894= 119904) [48] The filters
p119894and q
119894are optimized using the MMSE criterion and are
given in [6 24]For the first iteration since no a priori information is
available the equalization process is reduced to the classicalMMSE solution
119894= (H119867H +
1205902
119899
1205902119904
I119873119905
)
minus1
H119867y (23)
The equalized symbols 119894are associated with a bias factor 120573
119894
in addition to some residual noise plus interferences 120578119894
119894= 120573
119894119904119894+ 120578
119894 (24)
These equalized symbols are then used by the soft demapperto compute the LLR values using the max-log-MAP approxi-mation [48]
119871 (119909119894119887
) =1
1205902120578119894
(min119904119894isin120594minus1
119894119887
1003816100381610038161003816119894 minus 120573119894119904119894
1003816100381610038161003816
2
minus min119904119894isin120594+1
119894119887
1003816100381610038161003816119894 minus 120573119894119904119894
1003816100381610038161003816
2
) (25)
MMSE-IC equalizer requires 119873119905matrix inversions for each
symbol vector For this reason several approximations of
MMSE-IC were proposed For example in [9] a low-complexity approach of MMSE-IC is described by perform-ing a single matrix inversion without performance loss Thisalgorithm is referred to as LC-MMSE-IC
432 Successive Interference Cancellation (SIC) EqualizerThe SIC-based detector was initially used in the VBLASTsystems In VBLAST architecture [47] a successive cancel-lation step followed by an interference nulling step is usedto detect the transmitted symbols However this methodsuffers from error propagation An improved VBLAST foriterative detection and decoding is described in [49] At thefirst iteration an enhanced VBLAST which takes decisionerrors into account is employed [24] When the a prioriLLRs are available from the channel decoder soft symbols arecomputed by a soft mapper and are used in the interferencecancellation To describe the enhanced VBLAST algorithmwe assume that the detection order has been made accordingto the optimal detection order [47] For the 119894th step thepredetected symbol vector s
119894minus1until step 119894 minus 1 is canceled out
from the received signal
y119894= y minus H
1119894minus1s119894minus1
(26)
where s119894minus1
= [1
2
sdot sdot sdot 119894minus1
] and H1119894minus1
=
[h1h2
sdot sdot sdot h119894minus1
] with h119894being the 119894th column of H
Then the estimated symbol 119894is obtained using a filtered
matrix W119894based on the MMSE criterion that takes decision
errors into account [11 49]
119894= W119867
119894y119894= 120573
119894119904119894+ 120578
119894
W119894= 120590
2
119904(HΣ
119894H119867 + 119873
0I119873119903
)minus1
h119894
(27)
Σ119894is the decision error covariance matrix defined as
Σ119894=
119894minus1
sum
119895=1
1205982
119895e119895e119879119895
+
119873119905minus119894+1
sum
119895=119894
1205902
119904e119895e119879119895
1205982
119895= E
10038161003816100381610038161003816119904119895
minus 119895
10038161003816100381610038161003816
2
| 119895minus1
(28)
8 Mobile Information Systems
where e119894denotes a unit vector having zero components except
the 119894th component which is oneA soft demapper is then used to compute LLRs according
to (25) We refer to this algorithm as improved VBLAST (I-VBLAST) in the sequel
44 Low-Complexity K-Best Decoder The low-complexityK-Best (LC-K-Best) decoder recently proposed in [20] usestwo improvements over the classical K-Best decoder for thesake of lower complexity and latency The first improvementsimplifies the hybrid enumeration of the constellation pointsin real-valued systemmodel when the a priori information isincorporated into the tree search using two look-up tablesThe second improvement is to use a relaxed on-demandexpansion that reduces the need of exhaustive expansionand sorting operations The LC-K-Best algorithm can bedescribed as follows
The preprocessing step is as follows
(1) Input yH 119870CalculateH = QR y = Q119867y
(2) Enumerate the constellation symbols based on119898119860 for
all layers
The tree-search step is as follows
(1) Set layer to 2119873119905 listL = 0 119889
2119873119905+1
= 0
(a) expand all radic2119876 possible constellation nodes(b) calculate the corresponding PEDs(c) if radic2119876 gt 119870 select the 119870 best nodes and store
them in the listL2119873119905
(2) For layer 119894 = 2119873119905minus 1 1
(a) enumerate the constellation point according to119898119862
119894of the 119870 surviving paths in the listL
119894+1
(b) find the first child (FC) based on 119898119862
119894and 119898
119860
119894for
each parent nodes(c) compute their PEDs(d) select 119860 best children with smallest PEDs
among the 119870 FCs and add them to the listL119894
(e) if |L119894| lt 119870 find the next child (NC) of the
selected parent nodesCalculate their PEDs and go to step (2)(d)
(f) else move to the next layer 119894 = 119894 minus 1 and go tostep (2)
(3) If 119894 = 1 calculate the LLR as in (19)In the case of missing counter hypothesis LLR clip-ping method is used
It has been shown in [20] that the LC-K-Best decoderachieves almost the same performance as the classical K-Best decoder with different modulations Moreover thecomputational complexity in terms of the number of visitednodes is significantly reduced specially in the case of high-order modulations
5 Convergence of IterativeDetection-Decoding
The EXtrinsic Information Transfer (EXIT) chart is a usefultool to study the convergence behavior of iterative decod-ing systems [50] It describes the exchange of the mutualinformation in the iterative process in order to predict therequired number of iterations the convergence threshold(corresponding to the start of the waterfall region) and theaverage decoding trajectory
In the iterative receiver considered in our study two itera-tive processes are performed one inside the channel decoder(turbo or LDPC) and the other between the MIMO detectorand the channel decoder For simplicity we separately studythe convergence of the channel decoding and the MIMOdetection We denote by 119868
1198601and 119868
1198602the a priorimutual input
information of the MIMO detector and the channel decoderrespectively and by 119868
1198641and 119868
1198642their corresponding extrinsic
mutual output informationThe mutual information 119868
119909(119868119860or 119868
119864) can be computed
through Monte Carlo simulation using the probability den-sity function 119901
119871119909
[50]
119868119909
=1
2sum
119909isinminus11
int
+infin
minusinfin
119901119871119909
(119871119909
| 119909)
sdot log2
2119901119871119909
(119871119909
| 119909)
119901119871119909
(119871119909
| minus1) 119901119871119909
(119871119909
| +1)119889119871
119909
(29)
A simple approximation of the mutual information is used inour analysis [51]
119868119909
asymp 1 minus1
119871119887
119871119887minus1
sum
119899=0
log2
(1 + exp (minus119909119871119909)) (30)
where 119871119887is the number of transmitted bits and 119871
119909is the LLR
associated with the bit 119909 isin minus1 +1The a priori information 119871
119860can be modeled by applying
an independentGaussian randomvariable 119899119860with zeromean
and variance 1205902
119860in conjunction with the known transmitted
information bits 119909 [50]
119871119860
= 120583119860
119909 + 119899119860
where 120583119860
=1205902
119860
2 (31)
For each given mutual information value 119868119860
isin [0 1] 1205902
119860can
be computed using the following equation [52]
120590119860
asymp (minus1
1198671
log2
(1 minus 11986811198673
119860))
121198672
(32)
where 1198671
= 03073 1198672
= 08935 and 1198673
= 11064At the beginning the a priori mutual information is as
follows 1198681198601
= 0 and 1198681198602
= 0 Then the extrinsic mutualinformation 119868
1198641of the MIMO detector becomes the a priori
mutual information 1198681198602
of the channel decoder and so onand so forth (ie 119868
1198641= 119868
1198602and 119868
1198642= 119868
1198601) For a successful
decoding there must be an open tunnel between the curves
Mobile Information Systems 9
Table 1 Simulation parameters for convergence analysis
MIMO system 4 times 4 spatial multiplexingModulation 16-QAM Gray mappingChannel Flat Rayleigh fading
DetectorSTS-SD 119871 clip = plusmn8
LC-119870-Best 119870 = 16 119871 clip = plusmn3
MMSE-IC
Channel decoder
LTE turbo code [13 15]119900
Block length 119870119887
= 1024 119877119888
= 12
LDPC code (IEEE 80211n)Codeword length 119873
119887= 1944 119877
119888= 12
Interleaver Random size = 1024 (turbo code case)
the exchange of extrinsic information can be visualized as aldquozigzagrdquo decoding trajectory in the EXIT chart
To visualize the exchange of extrinsic information of theiterative receiver we present the MIMO detector and thechannel decoder characteristics into a single chart For ourconvergence analysis a 4 times 4 MIMO system with 16-QAMconstellation turbo decoder and LDPCdecoder (119877
119888= 12) is
considered Table 1 summarizes the main parameters for theconvergence analysis
Figure 5 shows the extrinsic information transfer char-acteristics of MIMO detectors at different 119864
119887119873
0values
As the I-VBLAST detector performs successive interferencecancellation at the first iteration and parallel interference can-cellation of the soft estimated symbols for the rest iterationsit is less intuitive to present its convergence in the EXITchart Therefore the convergence analysis of VBLAST is notconsidered
It is obvious that the characteristics of the detectorsare shifted upward with the increase of 119864
119887119873
0 We show
that for low 119864119887119873
0(0 dB) and for low mutual information
(lt01) MMSE-IC performs better than LC-K-Best decoderHowever with larger mutual information its performance islower Moreover the mutual information of STS-SD is higherthan LC-K-Best decoder and MMSE-IC for different 119864
119887119873
0
For higher 119864119887119873
0(5 dB) MMSE-IC presents lower mutual
information than other decoders when 1198681198601
lt 09Figure 6 shows the EXIT chart for 119864
119887119873
0= 2 dB with
several MIMO detectors namely STS-SD LC-K-Best andMMSE-IC We note that the characteristic of the channeldecoder is independent of 119864
119887119873
0values It is obvious that
the extrinsic mutual information of the channel decoderincreases with the number of iterations We see that after 6 to8 iterations in the case of turbo decoder (Figure 6(a)) thereis no significant improvement on the mutual informationMeanwhile in the case of LDPC decoder in Figure 6(b) 20iterations are enough for LDPC decoder to converge
By comparing the characteristics of STS-SD LC-K-Bestdecoder andMMSE-IC equalizer with both coding schemeswe notice that STS-SD has a larger mutual informationat its output LC-K-Best decoder has slightly less mutualinformation than STS-SD MMSE-IC shows low mutualinformation levels at its output compared to other algorithms
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
STS-SDLC-K-Best
MMSE-IC
EbN0 = 0 2 5dB
5dB
2dB
0dBI E1I
A2
IA1 IE2
Figure 5 Extrinsic information transfer characteristics of MIMOdetectors (STS-SD LC-K-Best andMMSE-IC) at119864
119887119873
0= 0 2 5 dB
in a 4 times 4 MIMO system using 16-QAM
when 1198681198601
lt 09 while for 1198681198601
gt 09 the extrinsic mutualinformation is comparable to others
In the case of turbo decoder (Figure 6(a)) with 119868in =
8 3 outer iterations are sufficient for STS-SD to convergeat 119864
119887119873
0= 2 dB However the same performance can
be attained by performing 4 outer iterations with only 2inner iterations Similarly LC-K-Best decoder shows anequivalent performance but slightly higher119864
119887119873
0is required
The convergence speed of LC-K-Best decoder is a bit lowerthan STS-SD which requires more iterations to get the sameperformance The reason is mainly due to the unreliabilityof LLRs caused by the small list size (L = 16) In thecase of MMSE-IC the characteristic presents a lower mutualinformation than the LC-K-Best decoder when 119868
1198601lt 09
Therefore an equivalent performance can be obtained athigher 119864
119887119873
0or by performing more iterations
In a similar way we study the convergence of MIMOdetection algorithms with LDPC decoder Figure 6(b) showsthe EXIT chart at 119864
119887119873
0= 2 dBThe same conclusion can be
retrieved as in the case of turbo decoder We can see that at119864119887119873
0= 2 dB a clear tunnel is observed between the MIMO
detector and the channel decoder characteristics allowingiterations to bring improvement to the system SimilarlySTS-SD offers higher mutual information than LC-K-Bestdecoder andMMSE-IC equalizer which suggests its superiorsymbol detection performance
Additionally the average decoding trajectory resultingfrom the free-run iterative detection-decoding simulationsis illustrated in Figure 6 at 119864
119887119873
0= 2 dB with 119868out = 4
119868in = 2 and 119868out = 4 119868in = 5 in the case of turbo decoder andLDPC decoder respectively The decoding trajectory closely
10 Mobile Information Systems
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
TurboSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 2 6 8
(a)
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
LDPCSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 5 10 20
(b)
Figure 6 EXIT chart for MIMO detectors (STS-SD LC-K-Best and MMSE-IC) and channel decoder (a) LTE turbo decoder 119870119887
= 1024119877119888
= 12 and (b) LDPC decoder 119873119887
= 1944 119877119888
= 12 at 119864119887119873
0= 2 dB in a 4 times 4 MIMO system using 16-QAM
matches the characteristics in the case of STS-SD and LC-K-Best decoders The little difference from the characteristicsafter a few iterations is due to the correlation of extrinsicinformation caused by the limited interleaver depth In thecase of MMSE-IC the decoding trajectory diverges fromthe characteristics for high mutual information because theequalizer uses the a posteriori information to compute softsymbols instead of the extrinsic information
The best trade-off scheduling of the required numberof iterations is therefore 119868out iterations in the outer loopand a total of 8 iterations inside the turbo decoder and 20iterations inside the LDPC decoder distributed across these119868out iterations
6 Performance and Complexity Evaluation ofIterative Detection-Decoding
In this section we evaluate and compare the performance andthe complexity of differentMIMOdetectors namely STS-SDLC-K-Best decoder MMSE-IC and I-VBLAST equalizerswith different channel coding techniques (turbo LDPC)A detailed analysis of the performance and the complexitytrade-off ofMIMOdetection with LTE turbo decoder and 16-QAM modulation in a Rayleigh channel has been discussedin [24] Herein the performance and the complexity ofthe receiver with LDPC decoder are investigated Moreoverseveral modulations and coding schemes are consideredto quantify the gain achieved by such iterative receiver indifferent channel environments Consequently a comparative
study is conducted in iterative receiver with both codingschemes (turbo LDPC)
For the turbo code the rate 13 turbo encoder specifiedin 3GPP LTE with a block length 119870
119887= 1024 is used in the
simulations Puncturing is performed in the rate matchingmodule to achieve an arbitrary coding rate 119877
119888(eg 119877
119888= 12
34)Meanwhile the LDPC encoder specified in IEEE 80211nis consideredThe encoder is defined by a parity checkmatrixthat is formed out of square submatrices of sizes 27 54 or 81Herein the codeword length of size 119873
119887= 1944 with coding
rate of 119877119888
= 12 and 34 is considered
61 Performance Evaluation The simulations are first carriedout in Rayleigh fading channel to view general performanceof the iterative receivers Real channel models will be consid-ered to evaluate the performance in more realistic scenariosTherefore the 3GPP LTE-(A) channel environments withlow medium and large delay spread values and Dopplerfrequencies are considered The low spread channel is theExtended Pedestrian A (EPA) model which emulates theurban environment with small cell sizes (120591rms = 43 ns) Themedium spread channel (120591rms = 357 ns) is the ExtendedVehicular A (EVA) model The Extended Typical Urban(ETU) model is the large spread channel which has alarger excess delay (120591rms = 991 ns) and simulates extremeurban suburban and rural cases Table 2 summarizes thecharacteristic parameters of these channel environments Forall cases the channel is assumed to be perfectly known atthe receiver Table 3 gives the principle parameters of thesimulations The performance is measured in terms of bit
Mobile Information Systems 11
Table 2 Characteristic parameters of the investigated channelmodels
error rate (BER) with respect to signal-to-noise ratio (SNR)per bit 119864
119887119873
0
119864119887
1198730
=119864119904
1198730
+ 10 sdot log10
1
119877119888119876119873
119905
[dB] (33)
In our previous study [24] the performance of MIMOdetectors with LTE turbo decoder is evaluated in a Rayleighchannel with various outer and inner iterations It has beenshown that the performance is improved by about 15 dB at aBER level of 1 times 10
minus5 with 4 outer iterations It has been alsoshown that no significant improvement can be achieved after4 outer iterations this improvement is less than 02 dB with 8outer iterations
Similarly to turbo decoder we fix the number of inneriterations inside LDPC decoder to 20 while varying thenumber of outer iterations Figure 7 shows BER performanceof MIMO detectors with LDPC decoder in Rayleigh channelwith 119868in = 20 iterations and 119868out = 1 2 4 or 8 iterationsSTS-SD is used with a LLR clipping level of 8 which givesclose toMAPperformancewith considerable reduction in thecomplexity For LC-K-Best decoder a LLR clipping level of 3is used in the case of missing counter hypothesis We show
0 1 2 3 4 5
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
Iout = 4
Iout = 8
Iout = 2
Iout = 1
EbN0 (dB)minus1
Figure 7 BER performance of a 4 times 4 spatial multiplexing systemwith 16-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) in a Rayleigh fading channel 119868out = 1 2 4 8 119868in =
20 LDPC decoder 119873119887
= 1944 and 119877119888
= 12
that performance improvement of 15 dB is observed with 4outer iterations For 119868out = 8 iterations the improvementis less than 02 dB Therefore 119868out = 4 iterations will beconsidered in the sequel
Figure 8 shows the BER performance of 4 times 4 16-QAMsystem in a Rayleigh fading channel with 119868out = 4 and 119868in = 2
in the case of turbo decoder and 119868in = [3 4 6 7] in the caseof LDPC decoder The notation 119868in = [3 4 6 7] denotes that3 inner iterations are performed in the 1st outer iteration 4inner iterations in the 2nd outer iteration and so on Theperformances of STS-SD with 119868in = 8 and 119868in = 20 foreach outer iteration in the case of turbo decoder and LDPCdecoder are also plotted as a reference In the case of turbodecoder we show that performing 119868in = 8 and 119868out = 4
iterations does not bring significant improvement on theperformance compared to the case when 119868in = 2 and 119868out =
4 iterations are performed Similarly in the case of LDPCdecoder the performance of 119868in = 20 and 119868out = 4 iterationsis comparable to the performance of 119868in = [3 4 6 7] and119868out = 4 iterations Hence using a large number of iterationsdoes not seem to be efficientwhich proves the results obtainedin the convergence analysis of Section 5
By comparing the algorithms LC-K-Best decoder showsa degradation of less than 02 dB compared to STS-SD ata BER level of 2 times 10
minus5 However it outperforms MMSE-IC and I-VBLAST equalizer by about 02 dB at a BERlevel of 2 times 10
minus5 MMSE-IC and I-VBLAST show almostthe same performance In addition in the case of LDPCdecoder (Figure 8(b)) we notice that increasing the numberof inner iterations for each outer iteration 119868in = [3 4 6 7]
12 Mobile Information Systems
0 05 1 15 2 25 3
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 8
(a) LTE turbo decoder
BER
0 05 1 15 2 25 3
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 20
STS-SD Iin = 5
STS-SD
LC-K-BestI-VBLASTMMSE-IC
(b) LDPC decoder
Figure 8 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
shows slightly better performance than performing an equalnumber of iterations 119868in = 5 for each outer iteration
In the case of high-order modulation higher spectralefficiency can be achieved at a cost of increased symboldetection difficulty Figure 9 shows the BER performance of64-QAM with 119877
119888= 34 We see that LC-K-Best decoder
with a list size of 32 presents the similar performance as STS-SD However I-VLAST equalizer and MMSE-IC equalizerpresent degradation of more than 2 dB at a BER level of 1 times
10minus5 compared to LC-K-Best decoder Therefore LC-K-Best
decoder is more robust in the case of high-ordermodulationsand high coding rates The figure also shows that the BERperformance of LDPC decoder is almost identical to that ofthe turbo decoder
In order to summarize the performance of differentdetectors with different channel decoders we provide the119864119887119873
0values achieving a BER level of 1 times 10
minus5 in Table 4The values given in the parentheses of the table represent theperformance loss compared to STS-SD
Next we evaluate the performance of the iterative receiverin more realistic channel environments Figures 10 11 and 12show the BER performance of the detectors with the channeldecoders on EPA EVA and ETU channels receptivelySimilar behaviors can be observed with LTE turbo decoderand with LDPC decoder
In EPA channel (Figure 10) we see that LC-K-Bestdecoder achieves similar performance as STS-SD in the caseof 64-AQM and presents a degradation less than 02 dB in the
case of 16-QAM Meanwhile MMSE-IC presents significantperformance loss of more than 6 dB in the case of 64-QAMand 119877
119888= 34 With 16-QAM and 119877
119888= 12 the degradation
of MMSE-IC compared to LC-K-Best decoder is about 1 dBat a BER level of 1 times 10
minus4In EVA channel (Figure 11) the performance loss of
MMSE-IC compared to LC-K-Best decoder is reduced toapproximately 5 dB with 64-QAM and 05 dB with 16-QAMLC-K-Best decoder presents a degradation of about 01sim03 dB compared to STS-SD in the case of 16-QAM and 64-QAM
Similarly in ETU channel (Figure 12) MMSE-IC presentsa performance degradation compared to LC-K-Best decoderThis degradation is less than 4 dB in the case of 64-QAM andless than 05 dB in the case of 16-QAM We notice also thatthe LC-K-Best decoder is comparable to STS-SD in the caseof 64-QAM and has a degradation of 02 dB in the case of 16-QAM
Comparing the performance of the iterative receiver indifferent channels it can be seen that iterative receiverspresent the best performance in ETU channel compared toEPA and EVA channels This is due to the high diversity ofETU channel At a BER level of 1 times 10
minus4 the performancegain in ETU channel in the case of LTE turbo decoder is about08 dB 13 dB compared to EVA channel with 16-QAM and64-QAM respectively In the case of LDPC decoder this gainis 04 dB and 1 dB with 16-QAM and 64-QAM respectivelyHowever in EPA channel the performance gain in ETU
Mobile Information Systems 13
3 4 5 6 7 8 9 10 11 12 13
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
EbN0 (dB)
(a) LTE turbo decoder
3 4 5 6 7 8 9 10 11 12
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)
STS-SD Iin = 20
STS-SDLC-K-Best
I-VBLASTMMSE-IC
(b) LDPC decoder
Figure 9 BER performance of a 4 times 4 spatial multiplexing system with 64-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
0values achieving a BER level of 1times 10minus5 for different detectors and channel decoders (turbo LDPC) in 4times 4 spatialmultiplexing
system with 16-QAM 119877119888
= 12 and 64-QAM 119877119888
= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
STS-SD 25 dB 102 dB 27 dB 96 dBLC-119870-Best 26 dB (minus01) 102 dB (00) 28 dB (minus01) 96 dB (00)I-VBLAST 28 dB (minus03) 128 dB (minus26) 29 dB (minus02) 12 dB (minus24)MMSE-IC 28 dB (minus03) 128 dB (minus26) 30 dB (minus03) 12 dB (minus24)lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
channel in the case of turbo decoder or LDPC decoder ismore than 1 dB with 16-QAM and 64-QAM
Table 5 summarizes the 119864119887119873
0values achieving a BER
level of 1times10minus4 of different detectors combined with different
channel decoders andmodulation orders in different channelmodels The values given in the parentheses in the tablerepresent the performance loss compared to STS-SD Asindicated in Table 5 the iterative receivers with turbo decoderand LDPC decoder have comparable performance with acoding rate 119877
119888= 12 (16-QAM) However with 119877
119888= 34
(64-QAM) the receivers with LDPC decoder present slightlya better performance especially in ETU channel (06 dB)
From these results we show that the iterative receiversubstantially improves the performance of coded MIMOsystems either with turbo decoder or with LDPC decoderin Rayleigh channel (Figures 8 and 9) and in more realisticchannels (Figures 10 11 and 12) Moreover we show that
performing a large number of inner iterations does notbring significant improvement In addition we show thatthe LC-K-Best decoder achieves a good performance withdifferent modulations and channel coding schemes Thefigures suggest that the BER performance of the iterativereceiver with turbo decoder is almost comparable to that ofthe LDPC decoder It is therefore meaningful to evaluate thecomputational complexity of the iterative receivers with bothdecoding techniques as it will be discussed in the next section
62 Complexity Evaluation The computational complexityhas significant impact on the latency throughput and powerconsumption of the deviceTherefore the receiver algorithmsshould be optimized to achieve a good trade-off betweenperformance and cost In this part we evaluate the com-putational complexity of the iterative receiver in terms ofbasic operations such as addition subtractionmultiplication
14 Mobile Information Systems
1 3 5 7 9 11 13 15 17 19
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
(a) EPA turbo 119868out = 4 119868in = 2
BER
1 3 5 7 9 11 13 15 17 19
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12 64-QAM
Rc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EPA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 10 BER performance of a 4 times 4 spatial multiplexing with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) on EPA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944 119868out = 4
119868in = [3 4 6 7]
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
STS-SDLC-K-Best
MMSE-IC
16-QAMRc = 12
64-QAMRc = 34
(a) EVA turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EVA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 11 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Best and MMSE-IC) on EVA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944
and 119868out = 4 119868in = [3 4 6 7]
Mobile Information Systems 15
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(a) ETU turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) ETU LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 12 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SDLC-K-Best and MMSE-IC) on ETU channel model (a) LTE turbo decoder 119870
minus4 in LTE channel models for different detectors and channel decoders (turbo LDPC)in 4 times 4 spatial multiplexing system with 16-QAM 119877
119888= 12 and 64-QAM 119877
119888= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
EPASTS-SD 63 dB 140 dB 62 dB 138 dB
LC-119870-Best 65 dB (minus02) 139 dB (+01) 64 dB (minus02) 139 dB (minus01)MMSE-IC 74 dB (minus11) gt20 dB (gtminus6) 80 dB (minus18) gt20 dB (gtminus6)
EVASTS-SD 52 dB 143 dB 53 dB 134 dB
LC-119870-Best 54 dB (minus02) 144 dB (minus01) 56 dB (minus03) 137 dB (minus03)MMSE-IC 58 dB (minus06) 190 dB (minus47) 60 dB (minus07) 185 dB (minus51)
ETUSTS-SD 44 dB 130 dB 49 dB 124 dB
LC-119870-Best 46 dB (minus02) 130 dB (00) 49 dB (00) 124 dB (00)MMSE-IC 49 dB (minus05) 175 dB (minus35) 51 dB (minus02) 16 dB (minus36)
lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
division square root extraction maximization and look-up table check (which are denoted by ADD SUB MULDIV SQRT Max and LUT resp) To this end the completecomparison of the iterative receivers with both channeldecoders (turbo LDPC) and with several modulations andcoding rates is carried out
621 Complexity of Iterative Receiver The complexity ofiterative receiver depends on theMIMOdetector the channeldecoder and the number of innerouter iterationsThis com-plexity can be expressed by
sdot [119862det1 + (119868out minus 1) sdot 119862deti] (34)
where 119862det1 denotes the complexity of the first iterationof MIMO detection algorithm per symbol vector withouttaking into consideration the a priori information 119862detidenotes the complexity per iteration per symbol vector takinginto consideration the a priori information 119862dec denotesthe complexity of the channel decoder per iteration perinformation bit 119873bit is the number of information bit at theinput of the encoder 119873symb is the number of symbol vectors119873symb and 119873bit are linked by the following relation
119873symb =119873bit
119876119877119888119873119905
= 120572119873bit with 120572 =1
119876119877119888119873119905
(35)
where 119876 is the number of bits in the constellation symbol 119877119888
is the coding rate and119873119905is the number of transmit antennas
16 Mobile Information Systems
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
43 Interference-Cancellation-Based Detection Interference-cancellation-based detection can be carried out either in aparallel way as inMMSE-IC [8 9] or in a successive way as inVBLAST [47]
431 Minimum Mean Square Error-Interference Cancella-tion (MMSE-IC) Equalizer MMSE-IC equalizer can be per-formed using two filters [4]The first filter p
119894is applied to the
received vector y and the second filter q119894is applied to the
estimated vector s in order to cancel the interference fromother layers The equalized symbol
119894can be written as
119894= p119867
119894y minus q119867
119894s119894
with 119894 isin [1 119873119905] (22)
where s119894denotes the estimated vector given by the
previous iteration with the 119894th symbol omitteds119894
= [1 sdot sdot sdot 119894minus1
0 119894+1
sdot sdot sdot 119873119905]
119894is calculated by the
soft mapper as 119894
= E[119904119894] = sum
119904isin2119876 119904119875 (119904
119894= 119904) [48] The filters
p119894and q
119894are optimized using the MMSE criterion and are
given in [6 24]For the first iteration since no a priori information is
available the equalization process is reduced to the classicalMMSE solution
119894= (H119867H +
1205902
119899
1205902119904
I119873119905
)
minus1
H119867y (23)
The equalized symbols 119894are associated with a bias factor 120573
119894
in addition to some residual noise plus interferences 120578119894
119894= 120573
119894119904119894+ 120578
119894 (24)
These equalized symbols are then used by the soft demapperto compute the LLR values using the max-log-MAP approxi-mation [48]
119871 (119909119894119887
) =1
1205902120578119894
(min119904119894isin120594minus1
119894119887
1003816100381610038161003816119894 minus 120573119894119904119894
1003816100381610038161003816
2
minus min119904119894isin120594+1
119894119887
1003816100381610038161003816119894 minus 120573119894119904119894
1003816100381610038161003816
2
) (25)
MMSE-IC equalizer requires 119873119905matrix inversions for each
symbol vector For this reason several approximations of
MMSE-IC were proposed For example in [9] a low-complexity approach of MMSE-IC is described by perform-ing a single matrix inversion without performance loss Thisalgorithm is referred to as LC-MMSE-IC
432 Successive Interference Cancellation (SIC) EqualizerThe SIC-based detector was initially used in the VBLASTsystems In VBLAST architecture [47] a successive cancel-lation step followed by an interference nulling step is usedto detect the transmitted symbols However this methodsuffers from error propagation An improved VBLAST foriterative detection and decoding is described in [49] At thefirst iteration an enhanced VBLAST which takes decisionerrors into account is employed [24] When the a prioriLLRs are available from the channel decoder soft symbols arecomputed by a soft mapper and are used in the interferencecancellation To describe the enhanced VBLAST algorithmwe assume that the detection order has been made accordingto the optimal detection order [47] For the 119894th step thepredetected symbol vector s
119894minus1until step 119894 minus 1 is canceled out
from the received signal
y119894= y minus H
1119894minus1s119894minus1
(26)
where s119894minus1
= [1
2
sdot sdot sdot 119894minus1
] and H1119894minus1
=
[h1h2
sdot sdot sdot h119894minus1
] with h119894being the 119894th column of H
Then the estimated symbol 119894is obtained using a filtered
matrix W119894based on the MMSE criterion that takes decision
errors into account [11 49]
119894= W119867
119894y119894= 120573
119894119904119894+ 120578
119894
W119894= 120590
2
119904(HΣ
119894H119867 + 119873
0I119873119903
)minus1
h119894
(27)
Σ119894is the decision error covariance matrix defined as
Σ119894=
119894minus1
sum
119895=1
1205982
119895e119895e119879119895
+
119873119905minus119894+1
sum
119895=119894
1205902
119904e119895e119879119895
1205982
119895= E
10038161003816100381610038161003816119904119895
minus 119895
10038161003816100381610038161003816
2
| 119895minus1
(28)
8 Mobile Information Systems
where e119894denotes a unit vector having zero components except
the 119894th component which is oneA soft demapper is then used to compute LLRs according
to (25) We refer to this algorithm as improved VBLAST (I-VBLAST) in the sequel
44 Low-Complexity K-Best Decoder The low-complexityK-Best (LC-K-Best) decoder recently proposed in [20] usestwo improvements over the classical K-Best decoder for thesake of lower complexity and latency The first improvementsimplifies the hybrid enumeration of the constellation pointsin real-valued systemmodel when the a priori information isincorporated into the tree search using two look-up tablesThe second improvement is to use a relaxed on-demandexpansion that reduces the need of exhaustive expansionand sorting operations The LC-K-Best algorithm can bedescribed as follows
The preprocessing step is as follows
(1) Input yH 119870CalculateH = QR y = Q119867y
(2) Enumerate the constellation symbols based on119898119860 for
all layers
The tree-search step is as follows
(1) Set layer to 2119873119905 listL = 0 119889
2119873119905+1
= 0
(a) expand all radic2119876 possible constellation nodes(b) calculate the corresponding PEDs(c) if radic2119876 gt 119870 select the 119870 best nodes and store
them in the listL2119873119905
(2) For layer 119894 = 2119873119905minus 1 1
(a) enumerate the constellation point according to119898119862
119894of the 119870 surviving paths in the listL
119894+1
(b) find the first child (FC) based on 119898119862
119894and 119898
119860
119894for
each parent nodes(c) compute their PEDs(d) select 119860 best children with smallest PEDs
among the 119870 FCs and add them to the listL119894
(e) if |L119894| lt 119870 find the next child (NC) of the
selected parent nodesCalculate their PEDs and go to step (2)(d)
(f) else move to the next layer 119894 = 119894 minus 1 and go tostep (2)
(3) If 119894 = 1 calculate the LLR as in (19)In the case of missing counter hypothesis LLR clip-ping method is used
It has been shown in [20] that the LC-K-Best decoderachieves almost the same performance as the classical K-Best decoder with different modulations Moreover thecomputational complexity in terms of the number of visitednodes is significantly reduced specially in the case of high-order modulations
5 Convergence of IterativeDetection-Decoding
The EXtrinsic Information Transfer (EXIT) chart is a usefultool to study the convergence behavior of iterative decod-ing systems [50] It describes the exchange of the mutualinformation in the iterative process in order to predict therequired number of iterations the convergence threshold(corresponding to the start of the waterfall region) and theaverage decoding trajectory
In the iterative receiver considered in our study two itera-tive processes are performed one inside the channel decoder(turbo or LDPC) and the other between the MIMO detectorand the channel decoder For simplicity we separately studythe convergence of the channel decoding and the MIMOdetection We denote by 119868
1198601and 119868
1198602the a priorimutual input
information of the MIMO detector and the channel decoderrespectively and by 119868
1198641and 119868
1198642their corresponding extrinsic
mutual output informationThe mutual information 119868
119909(119868119860or 119868
119864) can be computed
through Monte Carlo simulation using the probability den-sity function 119901
119871119909
[50]
119868119909
=1
2sum
119909isinminus11
int
+infin
minusinfin
119901119871119909
(119871119909
| 119909)
sdot log2
2119901119871119909
(119871119909
| 119909)
119901119871119909
(119871119909
| minus1) 119901119871119909
(119871119909
| +1)119889119871
119909
(29)
A simple approximation of the mutual information is used inour analysis [51]
119868119909
asymp 1 minus1
119871119887
119871119887minus1
sum
119899=0
log2
(1 + exp (minus119909119871119909)) (30)
where 119871119887is the number of transmitted bits and 119871
119909is the LLR
associated with the bit 119909 isin minus1 +1The a priori information 119871
119860can be modeled by applying
an independentGaussian randomvariable 119899119860with zeromean
and variance 1205902
119860in conjunction with the known transmitted
information bits 119909 [50]
119871119860
= 120583119860
119909 + 119899119860
where 120583119860
=1205902
119860
2 (31)
For each given mutual information value 119868119860
isin [0 1] 1205902
119860can
be computed using the following equation [52]
120590119860
asymp (minus1
1198671
log2
(1 minus 11986811198673
119860))
121198672
(32)
where 1198671
= 03073 1198672
= 08935 and 1198673
= 11064At the beginning the a priori mutual information is as
follows 1198681198601
= 0 and 1198681198602
= 0 Then the extrinsic mutualinformation 119868
1198641of the MIMO detector becomes the a priori
mutual information 1198681198602
of the channel decoder and so onand so forth (ie 119868
1198641= 119868
1198602and 119868
1198642= 119868
1198601) For a successful
decoding there must be an open tunnel between the curves
Mobile Information Systems 9
Table 1 Simulation parameters for convergence analysis
MIMO system 4 times 4 spatial multiplexingModulation 16-QAM Gray mappingChannel Flat Rayleigh fading
DetectorSTS-SD 119871 clip = plusmn8
LC-119870-Best 119870 = 16 119871 clip = plusmn3
MMSE-IC
Channel decoder
LTE turbo code [13 15]119900
Block length 119870119887
= 1024 119877119888
= 12
LDPC code (IEEE 80211n)Codeword length 119873
119887= 1944 119877
119888= 12
Interleaver Random size = 1024 (turbo code case)
the exchange of extrinsic information can be visualized as aldquozigzagrdquo decoding trajectory in the EXIT chart
To visualize the exchange of extrinsic information of theiterative receiver we present the MIMO detector and thechannel decoder characteristics into a single chart For ourconvergence analysis a 4 times 4 MIMO system with 16-QAMconstellation turbo decoder and LDPCdecoder (119877
119888= 12) is
considered Table 1 summarizes the main parameters for theconvergence analysis
Figure 5 shows the extrinsic information transfer char-acteristics of MIMO detectors at different 119864
119887119873
0values
As the I-VBLAST detector performs successive interferencecancellation at the first iteration and parallel interference can-cellation of the soft estimated symbols for the rest iterationsit is less intuitive to present its convergence in the EXITchart Therefore the convergence analysis of VBLAST is notconsidered
It is obvious that the characteristics of the detectorsare shifted upward with the increase of 119864
119887119873
0 We show
that for low 119864119887119873
0(0 dB) and for low mutual information
(lt01) MMSE-IC performs better than LC-K-Best decoderHowever with larger mutual information its performance islower Moreover the mutual information of STS-SD is higherthan LC-K-Best decoder and MMSE-IC for different 119864
119887119873
0
For higher 119864119887119873
0(5 dB) MMSE-IC presents lower mutual
information than other decoders when 1198681198601
lt 09Figure 6 shows the EXIT chart for 119864
119887119873
0= 2 dB with
several MIMO detectors namely STS-SD LC-K-Best andMMSE-IC We note that the characteristic of the channeldecoder is independent of 119864
119887119873
0values It is obvious that
the extrinsic mutual information of the channel decoderincreases with the number of iterations We see that after 6 to8 iterations in the case of turbo decoder (Figure 6(a)) thereis no significant improvement on the mutual informationMeanwhile in the case of LDPC decoder in Figure 6(b) 20iterations are enough for LDPC decoder to converge
By comparing the characteristics of STS-SD LC-K-Bestdecoder andMMSE-IC equalizer with both coding schemeswe notice that STS-SD has a larger mutual informationat its output LC-K-Best decoder has slightly less mutualinformation than STS-SD MMSE-IC shows low mutualinformation levels at its output compared to other algorithms
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
STS-SDLC-K-Best
MMSE-IC
EbN0 = 0 2 5dB
5dB
2dB
0dBI E1I
A2
IA1 IE2
Figure 5 Extrinsic information transfer characteristics of MIMOdetectors (STS-SD LC-K-Best andMMSE-IC) at119864
119887119873
0= 0 2 5 dB
in a 4 times 4 MIMO system using 16-QAM
when 1198681198601
lt 09 while for 1198681198601
gt 09 the extrinsic mutualinformation is comparable to others
In the case of turbo decoder (Figure 6(a)) with 119868in =
8 3 outer iterations are sufficient for STS-SD to convergeat 119864
119887119873
0= 2 dB However the same performance can
be attained by performing 4 outer iterations with only 2inner iterations Similarly LC-K-Best decoder shows anequivalent performance but slightly higher119864
119887119873
0is required
The convergence speed of LC-K-Best decoder is a bit lowerthan STS-SD which requires more iterations to get the sameperformance The reason is mainly due to the unreliabilityof LLRs caused by the small list size (L = 16) In thecase of MMSE-IC the characteristic presents a lower mutualinformation than the LC-K-Best decoder when 119868
1198601lt 09
Therefore an equivalent performance can be obtained athigher 119864
119887119873
0or by performing more iterations
In a similar way we study the convergence of MIMOdetection algorithms with LDPC decoder Figure 6(b) showsthe EXIT chart at 119864
119887119873
0= 2 dBThe same conclusion can be
retrieved as in the case of turbo decoder We can see that at119864119887119873
0= 2 dB a clear tunnel is observed between the MIMO
detector and the channel decoder characteristics allowingiterations to bring improvement to the system SimilarlySTS-SD offers higher mutual information than LC-K-Bestdecoder andMMSE-IC equalizer which suggests its superiorsymbol detection performance
Additionally the average decoding trajectory resultingfrom the free-run iterative detection-decoding simulationsis illustrated in Figure 6 at 119864
119887119873
0= 2 dB with 119868out = 4
119868in = 2 and 119868out = 4 119868in = 5 in the case of turbo decoder andLDPC decoder respectively The decoding trajectory closely
10 Mobile Information Systems
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
TurboSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 2 6 8
(a)
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
LDPCSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 5 10 20
(b)
Figure 6 EXIT chart for MIMO detectors (STS-SD LC-K-Best and MMSE-IC) and channel decoder (a) LTE turbo decoder 119870119887
= 1024119877119888
= 12 and (b) LDPC decoder 119873119887
= 1944 119877119888
= 12 at 119864119887119873
0= 2 dB in a 4 times 4 MIMO system using 16-QAM
matches the characteristics in the case of STS-SD and LC-K-Best decoders The little difference from the characteristicsafter a few iterations is due to the correlation of extrinsicinformation caused by the limited interleaver depth In thecase of MMSE-IC the decoding trajectory diverges fromthe characteristics for high mutual information because theequalizer uses the a posteriori information to compute softsymbols instead of the extrinsic information
The best trade-off scheduling of the required numberof iterations is therefore 119868out iterations in the outer loopand a total of 8 iterations inside the turbo decoder and 20iterations inside the LDPC decoder distributed across these119868out iterations
6 Performance and Complexity Evaluation ofIterative Detection-Decoding
In this section we evaluate and compare the performance andthe complexity of differentMIMOdetectors namely STS-SDLC-K-Best decoder MMSE-IC and I-VBLAST equalizerswith different channel coding techniques (turbo LDPC)A detailed analysis of the performance and the complexitytrade-off ofMIMOdetection with LTE turbo decoder and 16-QAM modulation in a Rayleigh channel has been discussedin [24] Herein the performance and the complexity ofthe receiver with LDPC decoder are investigated Moreoverseveral modulations and coding schemes are consideredto quantify the gain achieved by such iterative receiver indifferent channel environments Consequently a comparative
study is conducted in iterative receiver with both codingschemes (turbo LDPC)
For the turbo code the rate 13 turbo encoder specifiedin 3GPP LTE with a block length 119870
119887= 1024 is used in the
simulations Puncturing is performed in the rate matchingmodule to achieve an arbitrary coding rate 119877
119888(eg 119877
119888= 12
34)Meanwhile the LDPC encoder specified in IEEE 80211nis consideredThe encoder is defined by a parity checkmatrixthat is formed out of square submatrices of sizes 27 54 or 81Herein the codeword length of size 119873
119887= 1944 with coding
rate of 119877119888
= 12 and 34 is considered
61 Performance Evaluation The simulations are first carriedout in Rayleigh fading channel to view general performanceof the iterative receivers Real channel models will be consid-ered to evaluate the performance in more realistic scenariosTherefore the 3GPP LTE-(A) channel environments withlow medium and large delay spread values and Dopplerfrequencies are considered The low spread channel is theExtended Pedestrian A (EPA) model which emulates theurban environment with small cell sizes (120591rms = 43 ns) Themedium spread channel (120591rms = 357 ns) is the ExtendedVehicular A (EVA) model The Extended Typical Urban(ETU) model is the large spread channel which has alarger excess delay (120591rms = 991 ns) and simulates extremeurban suburban and rural cases Table 2 summarizes thecharacteristic parameters of these channel environments Forall cases the channel is assumed to be perfectly known atthe receiver Table 3 gives the principle parameters of thesimulations The performance is measured in terms of bit
Mobile Information Systems 11
Table 2 Characteristic parameters of the investigated channelmodels
error rate (BER) with respect to signal-to-noise ratio (SNR)per bit 119864
119887119873
0
119864119887
1198730
=119864119904
1198730
+ 10 sdot log10
1
119877119888119876119873
119905
[dB] (33)
In our previous study [24] the performance of MIMOdetectors with LTE turbo decoder is evaluated in a Rayleighchannel with various outer and inner iterations It has beenshown that the performance is improved by about 15 dB at aBER level of 1 times 10
minus5 with 4 outer iterations It has been alsoshown that no significant improvement can be achieved after4 outer iterations this improvement is less than 02 dB with 8outer iterations
Similarly to turbo decoder we fix the number of inneriterations inside LDPC decoder to 20 while varying thenumber of outer iterations Figure 7 shows BER performanceof MIMO detectors with LDPC decoder in Rayleigh channelwith 119868in = 20 iterations and 119868out = 1 2 4 or 8 iterationsSTS-SD is used with a LLR clipping level of 8 which givesclose toMAPperformancewith considerable reduction in thecomplexity For LC-K-Best decoder a LLR clipping level of 3is used in the case of missing counter hypothesis We show
0 1 2 3 4 5
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
Iout = 4
Iout = 8
Iout = 2
Iout = 1
EbN0 (dB)minus1
Figure 7 BER performance of a 4 times 4 spatial multiplexing systemwith 16-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) in a Rayleigh fading channel 119868out = 1 2 4 8 119868in =
20 LDPC decoder 119873119887
= 1944 and 119877119888
= 12
that performance improvement of 15 dB is observed with 4outer iterations For 119868out = 8 iterations the improvementis less than 02 dB Therefore 119868out = 4 iterations will beconsidered in the sequel
Figure 8 shows the BER performance of 4 times 4 16-QAMsystem in a Rayleigh fading channel with 119868out = 4 and 119868in = 2
in the case of turbo decoder and 119868in = [3 4 6 7] in the caseof LDPC decoder The notation 119868in = [3 4 6 7] denotes that3 inner iterations are performed in the 1st outer iteration 4inner iterations in the 2nd outer iteration and so on Theperformances of STS-SD with 119868in = 8 and 119868in = 20 foreach outer iteration in the case of turbo decoder and LDPCdecoder are also plotted as a reference In the case of turbodecoder we show that performing 119868in = 8 and 119868out = 4
iterations does not bring significant improvement on theperformance compared to the case when 119868in = 2 and 119868out =
4 iterations are performed Similarly in the case of LDPCdecoder the performance of 119868in = 20 and 119868out = 4 iterationsis comparable to the performance of 119868in = [3 4 6 7] and119868out = 4 iterations Hence using a large number of iterationsdoes not seem to be efficientwhich proves the results obtainedin the convergence analysis of Section 5
By comparing the algorithms LC-K-Best decoder showsa degradation of less than 02 dB compared to STS-SD ata BER level of 2 times 10
minus5 However it outperforms MMSE-IC and I-VBLAST equalizer by about 02 dB at a BERlevel of 2 times 10
minus5 MMSE-IC and I-VBLAST show almostthe same performance In addition in the case of LDPCdecoder (Figure 8(b)) we notice that increasing the numberof inner iterations for each outer iteration 119868in = [3 4 6 7]
12 Mobile Information Systems
0 05 1 15 2 25 3
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 8
(a) LTE turbo decoder
BER
0 05 1 15 2 25 3
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 20
STS-SD Iin = 5
STS-SD
LC-K-BestI-VBLASTMMSE-IC
(b) LDPC decoder
Figure 8 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
shows slightly better performance than performing an equalnumber of iterations 119868in = 5 for each outer iteration
In the case of high-order modulation higher spectralefficiency can be achieved at a cost of increased symboldetection difficulty Figure 9 shows the BER performance of64-QAM with 119877
119888= 34 We see that LC-K-Best decoder
with a list size of 32 presents the similar performance as STS-SD However I-VLAST equalizer and MMSE-IC equalizerpresent degradation of more than 2 dB at a BER level of 1 times
10minus5 compared to LC-K-Best decoder Therefore LC-K-Best
decoder is more robust in the case of high-ordermodulationsand high coding rates The figure also shows that the BERperformance of LDPC decoder is almost identical to that ofthe turbo decoder
In order to summarize the performance of differentdetectors with different channel decoders we provide the119864119887119873
0values achieving a BER level of 1 times 10
minus5 in Table 4The values given in the parentheses of the table represent theperformance loss compared to STS-SD
Next we evaluate the performance of the iterative receiverin more realistic channel environments Figures 10 11 and 12show the BER performance of the detectors with the channeldecoders on EPA EVA and ETU channels receptivelySimilar behaviors can be observed with LTE turbo decoderand with LDPC decoder
In EPA channel (Figure 10) we see that LC-K-Bestdecoder achieves similar performance as STS-SD in the caseof 64-AQM and presents a degradation less than 02 dB in the
case of 16-QAM Meanwhile MMSE-IC presents significantperformance loss of more than 6 dB in the case of 64-QAMand 119877
119888= 34 With 16-QAM and 119877
119888= 12 the degradation
of MMSE-IC compared to LC-K-Best decoder is about 1 dBat a BER level of 1 times 10
minus4In EVA channel (Figure 11) the performance loss of
MMSE-IC compared to LC-K-Best decoder is reduced toapproximately 5 dB with 64-QAM and 05 dB with 16-QAMLC-K-Best decoder presents a degradation of about 01sim03 dB compared to STS-SD in the case of 16-QAM and 64-QAM
Similarly in ETU channel (Figure 12) MMSE-IC presentsa performance degradation compared to LC-K-Best decoderThis degradation is less than 4 dB in the case of 64-QAM andless than 05 dB in the case of 16-QAM We notice also thatthe LC-K-Best decoder is comparable to STS-SD in the caseof 64-QAM and has a degradation of 02 dB in the case of 16-QAM
Comparing the performance of the iterative receiver indifferent channels it can be seen that iterative receiverspresent the best performance in ETU channel compared toEPA and EVA channels This is due to the high diversity ofETU channel At a BER level of 1 times 10
minus4 the performancegain in ETU channel in the case of LTE turbo decoder is about08 dB 13 dB compared to EVA channel with 16-QAM and64-QAM respectively In the case of LDPC decoder this gainis 04 dB and 1 dB with 16-QAM and 64-QAM respectivelyHowever in EPA channel the performance gain in ETU
Mobile Information Systems 13
3 4 5 6 7 8 9 10 11 12 13
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
EbN0 (dB)
(a) LTE turbo decoder
3 4 5 6 7 8 9 10 11 12
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)
STS-SD Iin = 20
STS-SDLC-K-Best
I-VBLASTMMSE-IC
(b) LDPC decoder
Figure 9 BER performance of a 4 times 4 spatial multiplexing system with 64-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
0values achieving a BER level of 1times 10minus5 for different detectors and channel decoders (turbo LDPC) in 4times 4 spatialmultiplexing
system with 16-QAM 119877119888
= 12 and 64-QAM 119877119888
= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
STS-SD 25 dB 102 dB 27 dB 96 dBLC-119870-Best 26 dB (minus01) 102 dB (00) 28 dB (minus01) 96 dB (00)I-VBLAST 28 dB (minus03) 128 dB (minus26) 29 dB (minus02) 12 dB (minus24)MMSE-IC 28 dB (minus03) 128 dB (minus26) 30 dB (minus03) 12 dB (minus24)lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
channel in the case of turbo decoder or LDPC decoder ismore than 1 dB with 16-QAM and 64-QAM
Table 5 summarizes the 119864119887119873
0values achieving a BER
level of 1times10minus4 of different detectors combined with different
channel decoders andmodulation orders in different channelmodels The values given in the parentheses in the tablerepresent the performance loss compared to STS-SD Asindicated in Table 5 the iterative receivers with turbo decoderand LDPC decoder have comparable performance with acoding rate 119877
119888= 12 (16-QAM) However with 119877
119888= 34
(64-QAM) the receivers with LDPC decoder present slightlya better performance especially in ETU channel (06 dB)
From these results we show that the iterative receiversubstantially improves the performance of coded MIMOsystems either with turbo decoder or with LDPC decoderin Rayleigh channel (Figures 8 and 9) and in more realisticchannels (Figures 10 11 and 12) Moreover we show that
performing a large number of inner iterations does notbring significant improvement In addition we show thatthe LC-K-Best decoder achieves a good performance withdifferent modulations and channel coding schemes Thefigures suggest that the BER performance of the iterativereceiver with turbo decoder is almost comparable to that ofthe LDPC decoder It is therefore meaningful to evaluate thecomputational complexity of the iterative receivers with bothdecoding techniques as it will be discussed in the next section
62 Complexity Evaluation The computational complexityhas significant impact on the latency throughput and powerconsumption of the deviceTherefore the receiver algorithmsshould be optimized to achieve a good trade-off betweenperformance and cost In this part we evaluate the com-putational complexity of the iterative receiver in terms ofbasic operations such as addition subtractionmultiplication
14 Mobile Information Systems
1 3 5 7 9 11 13 15 17 19
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
(a) EPA turbo 119868out = 4 119868in = 2
BER
1 3 5 7 9 11 13 15 17 19
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12 64-QAM
Rc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EPA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 10 BER performance of a 4 times 4 spatial multiplexing with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) on EPA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944 119868out = 4
119868in = [3 4 6 7]
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
STS-SDLC-K-Best
MMSE-IC
16-QAMRc = 12
64-QAMRc = 34
(a) EVA turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EVA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 11 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Best and MMSE-IC) on EVA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944
and 119868out = 4 119868in = [3 4 6 7]
Mobile Information Systems 15
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(a) ETU turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) ETU LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 12 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SDLC-K-Best and MMSE-IC) on ETU channel model (a) LTE turbo decoder 119870
minus4 in LTE channel models for different detectors and channel decoders (turbo LDPC)in 4 times 4 spatial multiplexing system with 16-QAM 119877
119888= 12 and 64-QAM 119877
119888= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
EPASTS-SD 63 dB 140 dB 62 dB 138 dB
LC-119870-Best 65 dB (minus02) 139 dB (+01) 64 dB (minus02) 139 dB (minus01)MMSE-IC 74 dB (minus11) gt20 dB (gtminus6) 80 dB (minus18) gt20 dB (gtminus6)
EVASTS-SD 52 dB 143 dB 53 dB 134 dB
LC-119870-Best 54 dB (minus02) 144 dB (minus01) 56 dB (minus03) 137 dB (minus03)MMSE-IC 58 dB (minus06) 190 dB (minus47) 60 dB (minus07) 185 dB (minus51)
ETUSTS-SD 44 dB 130 dB 49 dB 124 dB
LC-119870-Best 46 dB (minus02) 130 dB (00) 49 dB (00) 124 dB (00)MMSE-IC 49 dB (minus05) 175 dB (minus35) 51 dB (minus02) 16 dB (minus36)
lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
division square root extraction maximization and look-up table check (which are denoted by ADD SUB MULDIV SQRT Max and LUT resp) To this end the completecomparison of the iterative receivers with both channeldecoders (turbo LDPC) and with several modulations andcoding rates is carried out
621 Complexity of Iterative Receiver The complexity ofiterative receiver depends on theMIMOdetector the channeldecoder and the number of innerouter iterationsThis com-plexity can be expressed by
sdot [119862det1 + (119868out minus 1) sdot 119862deti] (34)
where 119862det1 denotes the complexity of the first iterationof MIMO detection algorithm per symbol vector withouttaking into consideration the a priori information 119862detidenotes the complexity per iteration per symbol vector takinginto consideration the a priori information 119862dec denotesthe complexity of the channel decoder per iteration perinformation bit 119873bit is the number of information bit at theinput of the encoder 119873symb is the number of symbol vectors119873symb and 119873bit are linked by the following relation
119873symb =119873bit
119876119877119888119873119905
= 120572119873bit with 120572 =1
119876119877119888119873119905
(35)
where 119876 is the number of bits in the constellation symbol 119877119888
is the coding rate and119873119905is the number of transmit antennas
16 Mobile Information Systems
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
where e119894denotes a unit vector having zero components except
the 119894th component which is oneA soft demapper is then used to compute LLRs according
to (25) We refer to this algorithm as improved VBLAST (I-VBLAST) in the sequel
44 Low-Complexity K-Best Decoder The low-complexityK-Best (LC-K-Best) decoder recently proposed in [20] usestwo improvements over the classical K-Best decoder for thesake of lower complexity and latency The first improvementsimplifies the hybrid enumeration of the constellation pointsin real-valued systemmodel when the a priori information isincorporated into the tree search using two look-up tablesThe second improvement is to use a relaxed on-demandexpansion that reduces the need of exhaustive expansionand sorting operations The LC-K-Best algorithm can bedescribed as follows
The preprocessing step is as follows
(1) Input yH 119870CalculateH = QR y = Q119867y
(2) Enumerate the constellation symbols based on119898119860 for
all layers
The tree-search step is as follows
(1) Set layer to 2119873119905 listL = 0 119889
2119873119905+1
= 0
(a) expand all radic2119876 possible constellation nodes(b) calculate the corresponding PEDs(c) if radic2119876 gt 119870 select the 119870 best nodes and store
them in the listL2119873119905
(2) For layer 119894 = 2119873119905minus 1 1
(a) enumerate the constellation point according to119898119862
119894of the 119870 surviving paths in the listL
119894+1
(b) find the first child (FC) based on 119898119862
119894and 119898
119860
119894for
each parent nodes(c) compute their PEDs(d) select 119860 best children with smallest PEDs
among the 119870 FCs and add them to the listL119894
(e) if |L119894| lt 119870 find the next child (NC) of the
selected parent nodesCalculate their PEDs and go to step (2)(d)
(f) else move to the next layer 119894 = 119894 minus 1 and go tostep (2)
(3) If 119894 = 1 calculate the LLR as in (19)In the case of missing counter hypothesis LLR clip-ping method is used
It has been shown in [20] that the LC-K-Best decoderachieves almost the same performance as the classical K-Best decoder with different modulations Moreover thecomputational complexity in terms of the number of visitednodes is significantly reduced specially in the case of high-order modulations
5 Convergence of IterativeDetection-Decoding
The EXtrinsic Information Transfer (EXIT) chart is a usefultool to study the convergence behavior of iterative decod-ing systems [50] It describes the exchange of the mutualinformation in the iterative process in order to predict therequired number of iterations the convergence threshold(corresponding to the start of the waterfall region) and theaverage decoding trajectory
In the iterative receiver considered in our study two itera-tive processes are performed one inside the channel decoder(turbo or LDPC) and the other between the MIMO detectorand the channel decoder For simplicity we separately studythe convergence of the channel decoding and the MIMOdetection We denote by 119868
1198601and 119868
1198602the a priorimutual input
information of the MIMO detector and the channel decoderrespectively and by 119868
1198641and 119868
1198642their corresponding extrinsic
mutual output informationThe mutual information 119868
119909(119868119860or 119868
119864) can be computed
through Monte Carlo simulation using the probability den-sity function 119901
119871119909
[50]
119868119909
=1
2sum
119909isinminus11
int
+infin
minusinfin
119901119871119909
(119871119909
| 119909)
sdot log2
2119901119871119909
(119871119909
| 119909)
119901119871119909
(119871119909
| minus1) 119901119871119909
(119871119909
| +1)119889119871
119909
(29)
A simple approximation of the mutual information is used inour analysis [51]
119868119909
asymp 1 minus1
119871119887
119871119887minus1
sum
119899=0
log2
(1 + exp (minus119909119871119909)) (30)
where 119871119887is the number of transmitted bits and 119871
119909is the LLR
associated with the bit 119909 isin minus1 +1The a priori information 119871
119860can be modeled by applying
an independentGaussian randomvariable 119899119860with zeromean
and variance 1205902
119860in conjunction with the known transmitted
information bits 119909 [50]
119871119860
= 120583119860
119909 + 119899119860
where 120583119860
=1205902
119860
2 (31)
For each given mutual information value 119868119860
isin [0 1] 1205902
119860can
be computed using the following equation [52]
120590119860
asymp (minus1
1198671
log2
(1 minus 11986811198673
119860))
121198672
(32)
where 1198671
= 03073 1198672
= 08935 and 1198673
= 11064At the beginning the a priori mutual information is as
follows 1198681198601
= 0 and 1198681198602
= 0 Then the extrinsic mutualinformation 119868
1198641of the MIMO detector becomes the a priori
mutual information 1198681198602
of the channel decoder and so onand so forth (ie 119868
1198641= 119868
1198602and 119868
1198642= 119868
1198601) For a successful
decoding there must be an open tunnel between the curves
Mobile Information Systems 9
Table 1 Simulation parameters for convergence analysis
MIMO system 4 times 4 spatial multiplexingModulation 16-QAM Gray mappingChannel Flat Rayleigh fading
DetectorSTS-SD 119871 clip = plusmn8
LC-119870-Best 119870 = 16 119871 clip = plusmn3
MMSE-IC
Channel decoder
LTE turbo code [13 15]119900
Block length 119870119887
= 1024 119877119888
= 12
LDPC code (IEEE 80211n)Codeword length 119873
119887= 1944 119877
119888= 12
Interleaver Random size = 1024 (turbo code case)
the exchange of extrinsic information can be visualized as aldquozigzagrdquo decoding trajectory in the EXIT chart
To visualize the exchange of extrinsic information of theiterative receiver we present the MIMO detector and thechannel decoder characteristics into a single chart For ourconvergence analysis a 4 times 4 MIMO system with 16-QAMconstellation turbo decoder and LDPCdecoder (119877
119888= 12) is
considered Table 1 summarizes the main parameters for theconvergence analysis
Figure 5 shows the extrinsic information transfer char-acteristics of MIMO detectors at different 119864
119887119873
0values
As the I-VBLAST detector performs successive interferencecancellation at the first iteration and parallel interference can-cellation of the soft estimated symbols for the rest iterationsit is less intuitive to present its convergence in the EXITchart Therefore the convergence analysis of VBLAST is notconsidered
It is obvious that the characteristics of the detectorsare shifted upward with the increase of 119864
119887119873
0 We show
that for low 119864119887119873
0(0 dB) and for low mutual information
(lt01) MMSE-IC performs better than LC-K-Best decoderHowever with larger mutual information its performance islower Moreover the mutual information of STS-SD is higherthan LC-K-Best decoder and MMSE-IC for different 119864
119887119873
0
For higher 119864119887119873
0(5 dB) MMSE-IC presents lower mutual
information than other decoders when 1198681198601
lt 09Figure 6 shows the EXIT chart for 119864
119887119873
0= 2 dB with
several MIMO detectors namely STS-SD LC-K-Best andMMSE-IC We note that the characteristic of the channeldecoder is independent of 119864
119887119873
0values It is obvious that
the extrinsic mutual information of the channel decoderincreases with the number of iterations We see that after 6 to8 iterations in the case of turbo decoder (Figure 6(a)) thereis no significant improvement on the mutual informationMeanwhile in the case of LDPC decoder in Figure 6(b) 20iterations are enough for LDPC decoder to converge
By comparing the characteristics of STS-SD LC-K-Bestdecoder andMMSE-IC equalizer with both coding schemeswe notice that STS-SD has a larger mutual informationat its output LC-K-Best decoder has slightly less mutualinformation than STS-SD MMSE-IC shows low mutualinformation levels at its output compared to other algorithms
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
STS-SDLC-K-Best
MMSE-IC
EbN0 = 0 2 5dB
5dB
2dB
0dBI E1I
A2
IA1 IE2
Figure 5 Extrinsic information transfer characteristics of MIMOdetectors (STS-SD LC-K-Best andMMSE-IC) at119864
119887119873
0= 0 2 5 dB
in a 4 times 4 MIMO system using 16-QAM
when 1198681198601
lt 09 while for 1198681198601
gt 09 the extrinsic mutualinformation is comparable to others
In the case of turbo decoder (Figure 6(a)) with 119868in =
8 3 outer iterations are sufficient for STS-SD to convergeat 119864
119887119873
0= 2 dB However the same performance can
be attained by performing 4 outer iterations with only 2inner iterations Similarly LC-K-Best decoder shows anequivalent performance but slightly higher119864
119887119873
0is required
The convergence speed of LC-K-Best decoder is a bit lowerthan STS-SD which requires more iterations to get the sameperformance The reason is mainly due to the unreliabilityof LLRs caused by the small list size (L = 16) In thecase of MMSE-IC the characteristic presents a lower mutualinformation than the LC-K-Best decoder when 119868
1198601lt 09
Therefore an equivalent performance can be obtained athigher 119864
119887119873
0or by performing more iterations
In a similar way we study the convergence of MIMOdetection algorithms with LDPC decoder Figure 6(b) showsthe EXIT chart at 119864
119887119873
0= 2 dBThe same conclusion can be
retrieved as in the case of turbo decoder We can see that at119864119887119873
0= 2 dB a clear tunnel is observed between the MIMO
detector and the channel decoder characteristics allowingiterations to bring improvement to the system SimilarlySTS-SD offers higher mutual information than LC-K-Bestdecoder andMMSE-IC equalizer which suggests its superiorsymbol detection performance
Additionally the average decoding trajectory resultingfrom the free-run iterative detection-decoding simulationsis illustrated in Figure 6 at 119864
119887119873
0= 2 dB with 119868out = 4
119868in = 2 and 119868out = 4 119868in = 5 in the case of turbo decoder andLDPC decoder respectively The decoding trajectory closely
10 Mobile Information Systems
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
TurboSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 2 6 8
(a)
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
LDPCSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 5 10 20
(b)
Figure 6 EXIT chart for MIMO detectors (STS-SD LC-K-Best and MMSE-IC) and channel decoder (a) LTE turbo decoder 119870119887
= 1024119877119888
= 12 and (b) LDPC decoder 119873119887
= 1944 119877119888
= 12 at 119864119887119873
0= 2 dB in a 4 times 4 MIMO system using 16-QAM
matches the characteristics in the case of STS-SD and LC-K-Best decoders The little difference from the characteristicsafter a few iterations is due to the correlation of extrinsicinformation caused by the limited interleaver depth In thecase of MMSE-IC the decoding trajectory diverges fromthe characteristics for high mutual information because theequalizer uses the a posteriori information to compute softsymbols instead of the extrinsic information
The best trade-off scheduling of the required numberof iterations is therefore 119868out iterations in the outer loopand a total of 8 iterations inside the turbo decoder and 20iterations inside the LDPC decoder distributed across these119868out iterations
6 Performance and Complexity Evaluation ofIterative Detection-Decoding
In this section we evaluate and compare the performance andthe complexity of differentMIMOdetectors namely STS-SDLC-K-Best decoder MMSE-IC and I-VBLAST equalizerswith different channel coding techniques (turbo LDPC)A detailed analysis of the performance and the complexitytrade-off ofMIMOdetection with LTE turbo decoder and 16-QAM modulation in a Rayleigh channel has been discussedin [24] Herein the performance and the complexity ofthe receiver with LDPC decoder are investigated Moreoverseveral modulations and coding schemes are consideredto quantify the gain achieved by such iterative receiver indifferent channel environments Consequently a comparative
study is conducted in iterative receiver with both codingschemes (turbo LDPC)
For the turbo code the rate 13 turbo encoder specifiedin 3GPP LTE with a block length 119870
119887= 1024 is used in the
simulations Puncturing is performed in the rate matchingmodule to achieve an arbitrary coding rate 119877
119888(eg 119877
119888= 12
34)Meanwhile the LDPC encoder specified in IEEE 80211nis consideredThe encoder is defined by a parity checkmatrixthat is formed out of square submatrices of sizes 27 54 or 81Herein the codeword length of size 119873
119887= 1944 with coding
rate of 119877119888
= 12 and 34 is considered
61 Performance Evaluation The simulations are first carriedout in Rayleigh fading channel to view general performanceof the iterative receivers Real channel models will be consid-ered to evaluate the performance in more realistic scenariosTherefore the 3GPP LTE-(A) channel environments withlow medium and large delay spread values and Dopplerfrequencies are considered The low spread channel is theExtended Pedestrian A (EPA) model which emulates theurban environment with small cell sizes (120591rms = 43 ns) Themedium spread channel (120591rms = 357 ns) is the ExtendedVehicular A (EVA) model The Extended Typical Urban(ETU) model is the large spread channel which has alarger excess delay (120591rms = 991 ns) and simulates extremeurban suburban and rural cases Table 2 summarizes thecharacteristic parameters of these channel environments Forall cases the channel is assumed to be perfectly known atthe receiver Table 3 gives the principle parameters of thesimulations The performance is measured in terms of bit
Mobile Information Systems 11
Table 2 Characteristic parameters of the investigated channelmodels
error rate (BER) with respect to signal-to-noise ratio (SNR)per bit 119864
119887119873
0
119864119887
1198730
=119864119904
1198730
+ 10 sdot log10
1
119877119888119876119873
119905
[dB] (33)
In our previous study [24] the performance of MIMOdetectors with LTE turbo decoder is evaluated in a Rayleighchannel with various outer and inner iterations It has beenshown that the performance is improved by about 15 dB at aBER level of 1 times 10
minus5 with 4 outer iterations It has been alsoshown that no significant improvement can be achieved after4 outer iterations this improvement is less than 02 dB with 8outer iterations
Similarly to turbo decoder we fix the number of inneriterations inside LDPC decoder to 20 while varying thenumber of outer iterations Figure 7 shows BER performanceof MIMO detectors with LDPC decoder in Rayleigh channelwith 119868in = 20 iterations and 119868out = 1 2 4 or 8 iterationsSTS-SD is used with a LLR clipping level of 8 which givesclose toMAPperformancewith considerable reduction in thecomplexity For LC-K-Best decoder a LLR clipping level of 3is used in the case of missing counter hypothesis We show
0 1 2 3 4 5
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
Iout = 4
Iout = 8
Iout = 2
Iout = 1
EbN0 (dB)minus1
Figure 7 BER performance of a 4 times 4 spatial multiplexing systemwith 16-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) in a Rayleigh fading channel 119868out = 1 2 4 8 119868in =
20 LDPC decoder 119873119887
= 1944 and 119877119888
= 12
that performance improvement of 15 dB is observed with 4outer iterations For 119868out = 8 iterations the improvementis less than 02 dB Therefore 119868out = 4 iterations will beconsidered in the sequel
Figure 8 shows the BER performance of 4 times 4 16-QAMsystem in a Rayleigh fading channel with 119868out = 4 and 119868in = 2
in the case of turbo decoder and 119868in = [3 4 6 7] in the caseof LDPC decoder The notation 119868in = [3 4 6 7] denotes that3 inner iterations are performed in the 1st outer iteration 4inner iterations in the 2nd outer iteration and so on Theperformances of STS-SD with 119868in = 8 and 119868in = 20 foreach outer iteration in the case of turbo decoder and LDPCdecoder are also plotted as a reference In the case of turbodecoder we show that performing 119868in = 8 and 119868out = 4
iterations does not bring significant improvement on theperformance compared to the case when 119868in = 2 and 119868out =
4 iterations are performed Similarly in the case of LDPCdecoder the performance of 119868in = 20 and 119868out = 4 iterationsis comparable to the performance of 119868in = [3 4 6 7] and119868out = 4 iterations Hence using a large number of iterationsdoes not seem to be efficientwhich proves the results obtainedin the convergence analysis of Section 5
By comparing the algorithms LC-K-Best decoder showsa degradation of less than 02 dB compared to STS-SD ata BER level of 2 times 10
minus5 However it outperforms MMSE-IC and I-VBLAST equalizer by about 02 dB at a BERlevel of 2 times 10
minus5 MMSE-IC and I-VBLAST show almostthe same performance In addition in the case of LDPCdecoder (Figure 8(b)) we notice that increasing the numberof inner iterations for each outer iteration 119868in = [3 4 6 7]
12 Mobile Information Systems
0 05 1 15 2 25 3
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 8
(a) LTE turbo decoder
BER
0 05 1 15 2 25 3
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 20
STS-SD Iin = 5
STS-SD
LC-K-BestI-VBLASTMMSE-IC
(b) LDPC decoder
Figure 8 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
shows slightly better performance than performing an equalnumber of iterations 119868in = 5 for each outer iteration
In the case of high-order modulation higher spectralefficiency can be achieved at a cost of increased symboldetection difficulty Figure 9 shows the BER performance of64-QAM with 119877
119888= 34 We see that LC-K-Best decoder
with a list size of 32 presents the similar performance as STS-SD However I-VLAST equalizer and MMSE-IC equalizerpresent degradation of more than 2 dB at a BER level of 1 times
10minus5 compared to LC-K-Best decoder Therefore LC-K-Best
decoder is more robust in the case of high-ordermodulationsand high coding rates The figure also shows that the BERperformance of LDPC decoder is almost identical to that ofthe turbo decoder
In order to summarize the performance of differentdetectors with different channel decoders we provide the119864119887119873
0values achieving a BER level of 1 times 10
minus5 in Table 4The values given in the parentheses of the table represent theperformance loss compared to STS-SD
Next we evaluate the performance of the iterative receiverin more realistic channel environments Figures 10 11 and 12show the BER performance of the detectors with the channeldecoders on EPA EVA and ETU channels receptivelySimilar behaviors can be observed with LTE turbo decoderand with LDPC decoder
In EPA channel (Figure 10) we see that LC-K-Bestdecoder achieves similar performance as STS-SD in the caseof 64-AQM and presents a degradation less than 02 dB in the
case of 16-QAM Meanwhile MMSE-IC presents significantperformance loss of more than 6 dB in the case of 64-QAMand 119877
119888= 34 With 16-QAM and 119877
119888= 12 the degradation
of MMSE-IC compared to LC-K-Best decoder is about 1 dBat a BER level of 1 times 10
minus4In EVA channel (Figure 11) the performance loss of
MMSE-IC compared to LC-K-Best decoder is reduced toapproximately 5 dB with 64-QAM and 05 dB with 16-QAMLC-K-Best decoder presents a degradation of about 01sim03 dB compared to STS-SD in the case of 16-QAM and 64-QAM
Similarly in ETU channel (Figure 12) MMSE-IC presentsa performance degradation compared to LC-K-Best decoderThis degradation is less than 4 dB in the case of 64-QAM andless than 05 dB in the case of 16-QAM We notice also thatthe LC-K-Best decoder is comparable to STS-SD in the caseof 64-QAM and has a degradation of 02 dB in the case of 16-QAM
Comparing the performance of the iterative receiver indifferent channels it can be seen that iterative receiverspresent the best performance in ETU channel compared toEPA and EVA channels This is due to the high diversity ofETU channel At a BER level of 1 times 10
minus4 the performancegain in ETU channel in the case of LTE turbo decoder is about08 dB 13 dB compared to EVA channel with 16-QAM and64-QAM respectively In the case of LDPC decoder this gainis 04 dB and 1 dB with 16-QAM and 64-QAM respectivelyHowever in EPA channel the performance gain in ETU
Mobile Information Systems 13
3 4 5 6 7 8 9 10 11 12 13
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
EbN0 (dB)
(a) LTE turbo decoder
3 4 5 6 7 8 9 10 11 12
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)
STS-SD Iin = 20
STS-SDLC-K-Best
I-VBLASTMMSE-IC
(b) LDPC decoder
Figure 9 BER performance of a 4 times 4 spatial multiplexing system with 64-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
0values achieving a BER level of 1times 10minus5 for different detectors and channel decoders (turbo LDPC) in 4times 4 spatialmultiplexing
system with 16-QAM 119877119888
= 12 and 64-QAM 119877119888
= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
STS-SD 25 dB 102 dB 27 dB 96 dBLC-119870-Best 26 dB (minus01) 102 dB (00) 28 dB (minus01) 96 dB (00)I-VBLAST 28 dB (minus03) 128 dB (minus26) 29 dB (minus02) 12 dB (minus24)MMSE-IC 28 dB (minus03) 128 dB (minus26) 30 dB (minus03) 12 dB (minus24)lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
channel in the case of turbo decoder or LDPC decoder ismore than 1 dB with 16-QAM and 64-QAM
Table 5 summarizes the 119864119887119873
0values achieving a BER
level of 1times10minus4 of different detectors combined with different
channel decoders andmodulation orders in different channelmodels The values given in the parentheses in the tablerepresent the performance loss compared to STS-SD Asindicated in Table 5 the iterative receivers with turbo decoderand LDPC decoder have comparable performance with acoding rate 119877
119888= 12 (16-QAM) However with 119877
119888= 34
(64-QAM) the receivers with LDPC decoder present slightlya better performance especially in ETU channel (06 dB)
From these results we show that the iterative receiversubstantially improves the performance of coded MIMOsystems either with turbo decoder or with LDPC decoderin Rayleigh channel (Figures 8 and 9) and in more realisticchannels (Figures 10 11 and 12) Moreover we show that
performing a large number of inner iterations does notbring significant improvement In addition we show thatthe LC-K-Best decoder achieves a good performance withdifferent modulations and channel coding schemes Thefigures suggest that the BER performance of the iterativereceiver with turbo decoder is almost comparable to that ofthe LDPC decoder It is therefore meaningful to evaluate thecomputational complexity of the iterative receivers with bothdecoding techniques as it will be discussed in the next section
62 Complexity Evaluation The computational complexityhas significant impact on the latency throughput and powerconsumption of the deviceTherefore the receiver algorithmsshould be optimized to achieve a good trade-off betweenperformance and cost In this part we evaluate the com-putational complexity of the iterative receiver in terms ofbasic operations such as addition subtractionmultiplication
14 Mobile Information Systems
1 3 5 7 9 11 13 15 17 19
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
(a) EPA turbo 119868out = 4 119868in = 2
BER
1 3 5 7 9 11 13 15 17 19
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12 64-QAM
Rc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EPA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 10 BER performance of a 4 times 4 spatial multiplexing with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) on EPA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944 119868out = 4
119868in = [3 4 6 7]
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
STS-SDLC-K-Best
MMSE-IC
16-QAMRc = 12
64-QAMRc = 34
(a) EVA turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EVA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 11 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Best and MMSE-IC) on EVA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944
and 119868out = 4 119868in = [3 4 6 7]
Mobile Information Systems 15
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(a) ETU turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) ETU LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 12 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SDLC-K-Best and MMSE-IC) on ETU channel model (a) LTE turbo decoder 119870
minus4 in LTE channel models for different detectors and channel decoders (turbo LDPC)in 4 times 4 spatial multiplexing system with 16-QAM 119877
119888= 12 and 64-QAM 119877
119888= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
EPASTS-SD 63 dB 140 dB 62 dB 138 dB
LC-119870-Best 65 dB (minus02) 139 dB (+01) 64 dB (minus02) 139 dB (minus01)MMSE-IC 74 dB (minus11) gt20 dB (gtminus6) 80 dB (minus18) gt20 dB (gtminus6)
EVASTS-SD 52 dB 143 dB 53 dB 134 dB
LC-119870-Best 54 dB (minus02) 144 dB (minus01) 56 dB (minus03) 137 dB (minus03)MMSE-IC 58 dB (minus06) 190 dB (minus47) 60 dB (minus07) 185 dB (minus51)
ETUSTS-SD 44 dB 130 dB 49 dB 124 dB
LC-119870-Best 46 dB (minus02) 130 dB (00) 49 dB (00) 124 dB (00)MMSE-IC 49 dB (minus05) 175 dB (minus35) 51 dB (minus02) 16 dB (minus36)
lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
division square root extraction maximization and look-up table check (which are denoted by ADD SUB MULDIV SQRT Max and LUT resp) To this end the completecomparison of the iterative receivers with both channeldecoders (turbo LDPC) and with several modulations andcoding rates is carried out
621 Complexity of Iterative Receiver The complexity ofiterative receiver depends on theMIMOdetector the channeldecoder and the number of innerouter iterationsThis com-plexity can be expressed by
sdot [119862det1 + (119868out minus 1) sdot 119862deti] (34)
where 119862det1 denotes the complexity of the first iterationof MIMO detection algorithm per symbol vector withouttaking into consideration the a priori information 119862detidenotes the complexity per iteration per symbol vector takinginto consideration the a priori information 119862dec denotesthe complexity of the channel decoder per iteration perinformation bit 119873bit is the number of information bit at theinput of the encoder 119873symb is the number of symbol vectors119873symb and 119873bit are linked by the following relation
119873symb =119873bit
119876119877119888119873119905
= 120572119873bit with 120572 =1
119876119877119888119873119905
(35)
where 119876 is the number of bits in the constellation symbol 119877119888
is the coding rate and119873119905is the number of transmit antennas
16 Mobile Information Systems
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
Table 1 Simulation parameters for convergence analysis
MIMO system 4 times 4 spatial multiplexingModulation 16-QAM Gray mappingChannel Flat Rayleigh fading
DetectorSTS-SD 119871 clip = plusmn8
LC-119870-Best 119870 = 16 119871 clip = plusmn3
MMSE-IC
Channel decoder
LTE turbo code [13 15]119900
Block length 119870119887
= 1024 119877119888
= 12
LDPC code (IEEE 80211n)Codeword length 119873
119887= 1944 119877
119888= 12
Interleaver Random size = 1024 (turbo code case)
the exchange of extrinsic information can be visualized as aldquozigzagrdquo decoding trajectory in the EXIT chart
To visualize the exchange of extrinsic information of theiterative receiver we present the MIMO detector and thechannel decoder characteristics into a single chart For ourconvergence analysis a 4 times 4 MIMO system with 16-QAMconstellation turbo decoder and LDPCdecoder (119877
119888= 12) is
considered Table 1 summarizes the main parameters for theconvergence analysis
Figure 5 shows the extrinsic information transfer char-acteristics of MIMO detectors at different 119864
119887119873
0values
As the I-VBLAST detector performs successive interferencecancellation at the first iteration and parallel interference can-cellation of the soft estimated symbols for the rest iterationsit is less intuitive to present its convergence in the EXITchart Therefore the convergence analysis of VBLAST is notconsidered
It is obvious that the characteristics of the detectorsare shifted upward with the increase of 119864
119887119873
0 We show
that for low 119864119887119873
0(0 dB) and for low mutual information
(lt01) MMSE-IC performs better than LC-K-Best decoderHowever with larger mutual information its performance islower Moreover the mutual information of STS-SD is higherthan LC-K-Best decoder and MMSE-IC for different 119864
119887119873
0
For higher 119864119887119873
0(5 dB) MMSE-IC presents lower mutual
information than other decoders when 1198681198601
lt 09Figure 6 shows the EXIT chart for 119864
119887119873
0= 2 dB with
several MIMO detectors namely STS-SD LC-K-Best andMMSE-IC We note that the characteristic of the channeldecoder is independent of 119864
119887119873
0values It is obvious that
the extrinsic mutual information of the channel decoderincreases with the number of iterations We see that after 6 to8 iterations in the case of turbo decoder (Figure 6(a)) thereis no significant improvement on the mutual informationMeanwhile in the case of LDPC decoder in Figure 6(b) 20iterations are enough for LDPC decoder to converge
By comparing the characteristics of STS-SD LC-K-Bestdecoder andMMSE-IC equalizer with both coding schemeswe notice that STS-SD has a larger mutual informationat its output LC-K-Best decoder has slightly less mutualinformation than STS-SD MMSE-IC shows low mutualinformation levels at its output compared to other algorithms
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
STS-SDLC-K-Best
MMSE-IC
EbN0 = 0 2 5dB
5dB
2dB
0dBI E1I
A2
IA1 IE2
Figure 5 Extrinsic information transfer characteristics of MIMOdetectors (STS-SD LC-K-Best andMMSE-IC) at119864
119887119873
0= 0 2 5 dB
in a 4 times 4 MIMO system using 16-QAM
when 1198681198601
lt 09 while for 1198681198601
gt 09 the extrinsic mutualinformation is comparable to others
In the case of turbo decoder (Figure 6(a)) with 119868in =
8 3 outer iterations are sufficient for STS-SD to convergeat 119864
119887119873
0= 2 dB However the same performance can
be attained by performing 4 outer iterations with only 2inner iterations Similarly LC-K-Best decoder shows anequivalent performance but slightly higher119864
119887119873
0is required
The convergence speed of LC-K-Best decoder is a bit lowerthan STS-SD which requires more iterations to get the sameperformance The reason is mainly due to the unreliabilityof LLRs caused by the small list size (L = 16) In thecase of MMSE-IC the characteristic presents a lower mutualinformation than the LC-K-Best decoder when 119868
1198601lt 09
Therefore an equivalent performance can be obtained athigher 119864
119887119873
0or by performing more iterations
In a similar way we study the convergence of MIMOdetection algorithms with LDPC decoder Figure 6(b) showsthe EXIT chart at 119864
119887119873
0= 2 dBThe same conclusion can be
retrieved as in the case of turbo decoder We can see that at119864119887119873
0= 2 dB a clear tunnel is observed between the MIMO
detector and the channel decoder characteristics allowingiterations to bring improvement to the system SimilarlySTS-SD offers higher mutual information than LC-K-Bestdecoder andMMSE-IC equalizer which suggests its superiorsymbol detection performance
Additionally the average decoding trajectory resultingfrom the free-run iterative detection-decoding simulationsis illustrated in Figure 6 at 119864
119887119873
0= 2 dB with 119868out = 4
119868in = 2 and 119868out = 4 119868in = 5 in the case of turbo decoder andLDPC decoder respectively The decoding trajectory closely
10 Mobile Information Systems
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
TurboSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 2 6 8
(a)
0 01 02 03 04 05 06 07 08 09 10
01
02
03
04
05
06
07
08
09
1
LDPCSTS-SD
LC-K-BestMMSE-IC
I E1I
A2
IA1 IE2
EbN0 = 2dB
Iin = 5 10 20
(b)
Figure 6 EXIT chart for MIMO detectors (STS-SD LC-K-Best and MMSE-IC) and channel decoder (a) LTE turbo decoder 119870119887
= 1024119877119888
= 12 and (b) LDPC decoder 119873119887
= 1944 119877119888
= 12 at 119864119887119873
0= 2 dB in a 4 times 4 MIMO system using 16-QAM
matches the characteristics in the case of STS-SD and LC-K-Best decoders The little difference from the characteristicsafter a few iterations is due to the correlation of extrinsicinformation caused by the limited interleaver depth In thecase of MMSE-IC the decoding trajectory diverges fromthe characteristics for high mutual information because theequalizer uses the a posteriori information to compute softsymbols instead of the extrinsic information
The best trade-off scheduling of the required numberof iterations is therefore 119868out iterations in the outer loopand a total of 8 iterations inside the turbo decoder and 20iterations inside the LDPC decoder distributed across these119868out iterations
6 Performance and Complexity Evaluation ofIterative Detection-Decoding
In this section we evaluate and compare the performance andthe complexity of differentMIMOdetectors namely STS-SDLC-K-Best decoder MMSE-IC and I-VBLAST equalizerswith different channel coding techniques (turbo LDPC)A detailed analysis of the performance and the complexitytrade-off ofMIMOdetection with LTE turbo decoder and 16-QAM modulation in a Rayleigh channel has been discussedin [24] Herein the performance and the complexity ofthe receiver with LDPC decoder are investigated Moreoverseveral modulations and coding schemes are consideredto quantify the gain achieved by such iterative receiver indifferent channel environments Consequently a comparative
study is conducted in iterative receiver with both codingschemes (turbo LDPC)
For the turbo code the rate 13 turbo encoder specifiedin 3GPP LTE with a block length 119870
119887= 1024 is used in the
simulations Puncturing is performed in the rate matchingmodule to achieve an arbitrary coding rate 119877
119888(eg 119877
119888= 12
34)Meanwhile the LDPC encoder specified in IEEE 80211nis consideredThe encoder is defined by a parity checkmatrixthat is formed out of square submatrices of sizes 27 54 or 81Herein the codeword length of size 119873
119887= 1944 with coding
rate of 119877119888
= 12 and 34 is considered
61 Performance Evaluation The simulations are first carriedout in Rayleigh fading channel to view general performanceof the iterative receivers Real channel models will be consid-ered to evaluate the performance in more realistic scenariosTherefore the 3GPP LTE-(A) channel environments withlow medium and large delay spread values and Dopplerfrequencies are considered The low spread channel is theExtended Pedestrian A (EPA) model which emulates theurban environment with small cell sizes (120591rms = 43 ns) Themedium spread channel (120591rms = 357 ns) is the ExtendedVehicular A (EVA) model The Extended Typical Urban(ETU) model is the large spread channel which has alarger excess delay (120591rms = 991 ns) and simulates extremeurban suburban and rural cases Table 2 summarizes thecharacteristic parameters of these channel environments Forall cases the channel is assumed to be perfectly known atthe receiver Table 3 gives the principle parameters of thesimulations The performance is measured in terms of bit
Mobile Information Systems 11
Table 2 Characteristic parameters of the investigated channelmodels
error rate (BER) with respect to signal-to-noise ratio (SNR)per bit 119864
119887119873
0
119864119887
1198730
=119864119904
1198730
+ 10 sdot log10
1
119877119888119876119873
119905
[dB] (33)
In our previous study [24] the performance of MIMOdetectors with LTE turbo decoder is evaluated in a Rayleighchannel with various outer and inner iterations It has beenshown that the performance is improved by about 15 dB at aBER level of 1 times 10
minus5 with 4 outer iterations It has been alsoshown that no significant improvement can be achieved after4 outer iterations this improvement is less than 02 dB with 8outer iterations
Similarly to turbo decoder we fix the number of inneriterations inside LDPC decoder to 20 while varying thenumber of outer iterations Figure 7 shows BER performanceof MIMO detectors with LDPC decoder in Rayleigh channelwith 119868in = 20 iterations and 119868out = 1 2 4 or 8 iterationsSTS-SD is used with a LLR clipping level of 8 which givesclose toMAPperformancewith considerable reduction in thecomplexity For LC-K-Best decoder a LLR clipping level of 3is used in the case of missing counter hypothesis We show
0 1 2 3 4 5
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
Iout = 4
Iout = 8
Iout = 2
Iout = 1
EbN0 (dB)minus1
Figure 7 BER performance of a 4 times 4 spatial multiplexing systemwith 16-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) in a Rayleigh fading channel 119868out = 1 2 4 8 119868in =
20 LDPC decoder 119873119887
= 1944 and 119877119888
= 12
that performance improvement of 15 dB is observed with 4outer iterations For 119868out = 8 iterations the improvementis less than 02 dB Therefore 119868out = 4 iterations will beconsidered in the sequel
Figure 8 shows the BER performance of 4 times 4 16-QAMsystem in a Rayleigh fading channel with 119868out = 4 and 119868in = 2
in the case of turbo decoder and 119868in = [3 4 6 7] in the caseof LDPC decoder The notation 119868in = [3 4 6 7] denotes that3 inner iterations are performed in the 1st outer iteration 4inner iterations in the 2nd outer iteration and so on Theperformances of STS-SD with 119868in = 8 and 119868in = 20 foreach outer iteration in the case of turbo decoder and LDPCdecoder are also plotted as a reference In the case of turbodecoder we show that performing 119868in = 8 and 119868out = 4
iterations does not bring significant improvement on theperformance compared to the case when 119868in = 2 and 119868out =
4 iterations are performed Similarly in the case of LDPCdecoder the performance of 119868in = 20 and 119868out = 4 iterationsis comparable to the performance of 119868in = [3 4 6 7] and119868out = 4 iterations Hence using a large number of iterationsdoes not seem to be efficientwhich proves the results obtainedin the convergence analysis of Section 5
By comparing the algorithms LC-K-Best decoder showsa degradation of less than 02 dB compared to STS-SD ata BER level of 2 times 10
minus5 However it outperforms MMSE-IC and I-VBLAST equalizer by about 02 dB at a BERlevel of 2 times 10
minus5 MMSE-IC and I-VBLAST show almostthe same performance In addition in the case of LDPCdecoder (Figure 8(b)) we notice that increasing the numberof inner iterations for each outer iteration 119868in = [3 4 6 7]
12 Mobile Information Systems
0 05 1 15 2 25 3
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 8
(a) LTE turbo decoder
BER
0 05 1 15 2 25 3
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 20
STS-SD Iin = 5
STS-SD
LC-K-BestI-VBLASTMMSE-IC
(b) LDPC decoder
Figure 8 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
shows slightly better performance than performing an equalnumber of iterations 119868in = 5 for each outer iteration
In the case of high-order modulation higher spectralefficiency can be achieved at a cost of increased symboldetection difficulty Figure 9 shows the BER performance of64-QAM with 119877
119888= 34 We see that LC-K-Best decoder
with a list size of 32 presents the similar performance as STS-SD However I-VLAST equalizer and MMSE-IC equalizerpresent degradation of more than 2 dB at a BER level of 1 times
10minus5 compared to LC-K-Best decoder Therefore LC-K-Best
decoder is more robust in the case of high-ordermodulationsand high coding rates The figure also shows that the BERperformance of LDPC decoder is almost identical to that ofthe turbo decoder
In order to summarize the performance of differentdetectors with different channel decoders we provide the119864119887119873
0values achieving a BER level of 1 times 10
minus5 in Table 4The values given in the parentheses of the table represent theperformance loss compared to STS-SD
Next we evaluate the performance of the iterative receiverin more realistic channel environments Figures 10 11 and 12show the BER performance of the detectors with the channeldecoders on EPA EVA and ETU channels receptivelySimilar behaviors can be observed with LTE turbo decoderand with LDPC decoder
In EPA channel (Figure 10) we see that LC-K-Bestdecoder achieves similar performance as STS-SD in the caseof 64-AQM and presents a degradation less than 02 dB in the
case of 16-QAM Meanwhile MMSE-IC presents significantperformance loss of more than 6 dB in the case of 64-QAMand 119877
119888= 34 With 16-QAM and 119877
119888= 12 the degradation
of MMSE-IC compared to LC-K-Best decoder is about 1 dBat a BER level of 1 times 10
minus4In EVA channel (Figure 11) the performance loss of
MMSE-IC compared to LC-K-Best decoder is reduced toapproximately 5 dB with 64-QAM and 05 dB with 16-QAMLC-K-Best decoder presents a degradation of about 01sim03 dB compared to STS-SD in the case of 16-QAM and 64-QAM
Similarly in ETU channel (Figure 12) MMSE-IC presentsa performance degradation compared to LC-K-Best decoderThis degradation is less than 4 dB in the case of 64-QAM andless than 05 dB in the case of 16-QAM We notice also thatthe LC-K-Best decoder is comparable to STS-SD in the caseof 64-QAM and has a degradation of 02 dB in the case of 16-QAM
Comparing the performance of the iterative receiver indifferent channels it can be seen that iterative receiverspresent the best performance in ETU channel compared toEPA and EVA channels This is due to the high diversity ofETU channel At a BER level of 1 times 10
minus4 the performancegain in ETU channel in the case of LTE turbo decoder is about08 dB 13 dB compared to EVA channel with 16-QAM and64-QAM respectively In the case of LDPC decoder this gainis 04 dB and 1 dB with 16-QAM and 64-QAM respectivelyHowever in EPA channel the performance gain in ETU
Mobile Information Systems 13
3 4 5 6 7 8 9 10 11 12 13
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
EbN0 (dB)
(a) LTE turbo decoder
3 4 5 6 7 8 9 10 11 12
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)
STS-SD Iin = 20
STS-SDLC-K-Best
I-VBLASTMMSE-IC
(b) LDPC decoder
Figure 9 BER performance of a 4 times 4 spatial multiplexing system with 64-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
0values achieving a BER level of 1times 10minus5 for different detectors and channel decoders (turbo LDPC) in 4times 4 spatialmultiplexing
system with 16-QAM 119877119888
= 12 and 64-QAM 119877119888
= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
STS-SD 25 dB 102 dB 27 dB 96 dBLC-119870-Best 26 dB (minus01) 102 dB (00) 28 dB (minus01) 96 dB (00)I-VBLAST 28 dB (minus03) 128 dB (minus26) 29 dB (minus02) 12 dB (minus24)MMSE-IC 28 dB (minus03) 128 dB (minus26) 30 dB (minus03) 12 dB (minus24)lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
channel in the case of turbo decoder or LDPC decoder ismore than 1 dB with 16-QAM and 64-QAM
Table 5 summarizes the 119864119887119873
0values achieving a BER
level of 1times10minus4 of different detectors combined with different
channel decoders andmodulation orders in different channelmodels The values given in the parentheses in the tablerepresent the performance loss compared to STS-SD Asindicated in Table 5 the iterative receivers with turbo decoderand LDPC decoder have comparable performance with acoding rate 119877
119888= 12 (16-QAM) However with 119877
119888= 34
(64-QAM) the receivers with LDPC decoder present slightlya better performance especially in ETU channel (06 dB)
From these results we show that the iterative receiversubstantially improves the performance of coded MIMOsystems either with turbo decoder or with LDPC decoderin Rayleigh channel (Figures 8 and 9) and in more realisticchannels (Figures 10 11 and 12) Moreover we show that
performing a large number of inner iterations does notbring significant improvement In addition we show thatthe LC-K-Best decoder achieves a good performance withdifferent modulations and channel coding schemes Thefigures suggest that the BER performance of the iterativereceiver with turbo decoder is almost comparable to that ofthe LDPC decoder It is therefore meaningful to evaluate thecomputational complexity of the iterative receivers with bothdecoding techniques as it will be discussed in the next section
62 Complexity Evaluation The computational complexityhas significant impact on the latency throughput and powerconsumption of the deviceTherefore the receiver algorithmsshould be optimized to achieve a good trade-off betweenperformance and cost In this part we evaluate the com-putational complexity of the iterative receiver in terms ofbasic operations such as addition subtractionmultiplication
14 Mobile Information Systems
1 3 5 7 9 11 13 15 17 19
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
(a) EPA turbo 119868out = 4 119868in = 2
BER
1 3 5 7 9 11 13 15 17 19
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12 64-QAM
Rc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EPA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 10 BER performance of a 4 times 4 spatial multiplexing with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) on EPA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944 119868out = 4
119868in = [3 4 6 7]
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
STS-SDLC-K-Best
MMSE-IC
16-QAMRc = 12
64-QAMRc = 34
(a) EVA turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EVA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 11 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Best and MMSE-IC) on EVA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944
and 119868out = 4 119868in = [3 4 6 7]
Mobile Information Systems 15
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(a) ETU turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) ETU LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 12 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SDLC-K-Best and MMSE-IC) on ETU channel model (a) LTE turbo decoder 119870
minus4 in LTE channel models for different detectors and channel decoders (turbo LDPC)in 4 times 4 spatial multiplexing system with 16-QAM 119877
119888= 12 and 64-QAM 119877
119888= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
EPASTS-SD 63 dB 140 dB 62 dB 138 dB
LC-119870-Best 65 dB (minus02) 139 dB (+01) 64 dB (minus02) 139 dB (minus01)MMSE-IC 74 dB (minus11) gt20 dB (gtminus6) 80 dB (minus18) gt20 dB (gtminus6)
EVASTS-SD 52 dB 143 dB 53 dB 134 dB
LC-119870-Best 54 dB (minus02) 144 dB (minus01) 56 dB (minus03) 137 dB (minus03)MMSE-IC 58 dB (minus06) 190 dB (minus47) 60 dB (minus07) 185 dB (minus51)
ETUSTS-SD 44 dB 130 dB 49 dB 124 dB
LC-119870-Best 46 dB (minus02) 130 dB (00) 49 dB (00) 124 dB (00)MMSE-IC 49 dB (minus05) 175 dB (minus35) 51 dB (minus02) 16 dB (minus36)
lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
division square root extraction maximization and look-up table check (which are denoted by ADD SUB MULDIV SQRT Max and LUT resp) To this end the completecomparison of the iterative receivers with both channeldecoders (turbo LDPC) and with several modulations andcoding rates is carried out
621 Complexity of Iterative Receiver The complexity ofiterative receiver depends on theMIMOdetector the channeldecoder and the number of innerouter iterationsThis com-plexity can be expressed by
sdot [119862det1 + (119868out minus 1) sdot 119862deti] (34)
where 119862det1 denotes the complexity of the first iterationof MIMO detection algorithm per symbol vector withouttaking into consideration the a priori information 119862detidenotes the complexity per iteration per symbol vector takinginto consideration the a priori information 119862dec denotesthe complexity of the channel decoder per iteration perinformation bit 119873bit is the number of information bit at theinput of the encoder 119873symb is the number of symbol vectors119873symb and 119873bit are linked by the following relation
119873symb =119873bit
119876119877119888119873119905
= 120572119873bit with 120572 =1
119876119877119888119873119905
(35)
where 119876 is the number of bits in the constellation symbol 119877119888
is the coding rate and119873119905is the number of transmit antennas
16 Mobile Information Systems
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
Figure 6 EXIT chart for MIMO detectors (STS-SD LC-K-Best and MMSE-IC) and channel decoder (a) LTE turbo decoder 119870119887
= 1024119877119888
= 12 and (b) LDPC decoder 119873119887
= 1944 119877119888
= 12 at 119864119887119873
0= 2 dB in a 4 times 4 MIMO system using 16-QAM
matches the characteristics in the case of STS-SD and LC-K-Best decoders The little difference from the characteristicsafter a few iterations is due to the correlation of extrinsicinformation caused by the limited interleaver depth In thecase of MMSE-IC the decoding trajectory diverges fromthe characteristics for high mutual information because theequalizer uses the a posteriori information to compute softsymbols instead of the extrinsic information
The best trade-off scheduling of the required numberof iterations is therefore 119868out iterations in the outer loopand a total of 8 iterations inside the turbo decoder and 20iterations inside the LDPC decoder distributed across these119868out iterations
6 Performance and Complexity Evaluation ofIterative Detection-Decoding
In this section we evaluate and compare the performance andthe complexity of differentMIMOdetectors namely STS-SDLC-K-Best decoder MMSE-IC and I-VBLAST equalizerswith different channel coding techniques (turbo LDPC)A detailed analysis of the performance and the complexitytrade-off ofMIMOdetection with LTE turbo decoder and 16-QAM modulation in a Rayleigh channel has been discussedin [24] Herein the performance and the complexity ofthe receiver with LDPC decoder are investigated Moreoverseveral modulations and coding schemes are consideredto quantify the gain achieved by such iterative receiver indifferent channel environments Consequently a comparative
study is conducted in iterative receiver with both codingschemes (turbo LDPC)
For the turbo code the rate 13 turbo encoder specifiedin 3GPP LTE with a block length 119870
119887= 1024 is used in the
simulations Puncturing is performed in the rate matchingmodule to achieve an arbitrary coding rate 119877
119888(eg 119877
119888= 12
34)Meanwhile the LDPC encoder specified in IEEE 80211nis consideredThe encoder is defined by a parity checkmatrixthat is formed out of square submatrices of sizes 27 54 or 81Herein the codeword length of size 119873
119887= 1944 with coding
rate of 119877119888
= 12 and 34 is considered
61 Performance Evaluation The simulations are first carriedout in Rayleigh fading channel to view general performanceof the iterative receivers Real channel models will be consid-ered to evaluate the performance in more realistic scenariosTherefore the 3GPP LTE-(A) channel environments withlow medium and large delay spread values and Dopplerfrequencies are considered The low spread channel is theExtended Pedestrian A (EPA) model which emulates theurban environment with small cell sizes (120591rms = 43 ns) Themedium spread channel (120591rms = 357 ns) is the ExtendedVehicular A (EVA) model The Extended Typical Urban(ETU) model is the large spread channel which has alarger excess delay (120591rms = 991 ns) and simulates extremeurban suburban and rural cases Table 2 summarizes thecharacteristic parameters of these channel environments Forall cases the channel is assumed to be perfectly known atthe receiver Table 3 gives the principle parameters of thesimulations The performance is measured in terms of bit
Mobile Information Systems 11
Table 2 Characteristic parameters of the investigated channelmodels
error rate (BER) with respect to signal-to-noise ratio (SNR)per bit 119864
119887119873
0
119864119887
1198730
=119864119904
1198730
+ 10 sdot log10
1
119877119888119876119873
119905
[dB] (33)
In our previous study [24] the performance of MIMOdetectors with LTE turbo decoder is evaluated in a Rayleighchannel with various outer and inner iterations It has beenshown that the performance is improved by about 15 dB at aBER level of 1 times 10
minus5 with 4 outer iterations It has been alsoshown that no significant improvement can be achieved after4 outer iterations this improvement is less than 02 dB with 8outer iterations
Similarly to turbo decoder we fix the number of inneriterations inside LDPC decoder to 20 while varying thenumber of outer iterations Figure 7 shows BER performanceof MIMO detectors with LDPC decoder in Rayleigh channelwith 119868in = 20 iterations and 119868out = 1 2 4 or 8 iterationsSTS-SD is used with a LLR clipping level of 8 which givesclose toMAPperformancewith considerable reduction in thecomplexity For LC-K-Best decoder a LLR clipping level of 3is used in the case of missing counter hypothesis We show
0 1 2 3 4 5
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
Iout = 4
Iout = 8
Iout = 2
Iout = 1
EbN0 (dB)minus1
Figure 7 BER performance of a 4 times 4 spatial multiplexing systemwith 16-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) in a Rayleigh fading channel 119868out = 1 2 4 8 119868in =
20 LDPC decoder 119873119887
= 1944 and 119877119888
= 12
that performance improvement of 15 dB is observed with 4outer iterations For 119868out = 8 iterations the improvementis less than 02 dB Therefore 119868out = 4 iterations will beconsidered in the sequel
Figure 8 shows the BER performance of 4 times 4 16-QAMsystem in a Rayleigh fading channel with 119868out = 4 and 119868in = 2
in the case of turbo decoder and 119868in = [3 4 6 7] in the caseof LDPC decoder The notation 119868in = [3 4 6 7] denotes that3 inner iterations are performed in the 1st outer iteration 4inner iterations in the 2nd outer iteration and so on Theperformances of STS-SD with 119868in = 8 and 119868in = 20 foreach outer iteration in the case of turbo decoder and LDPCdecoder are also plotted as a reference In the case of turbodecoder we show that performing 119868in = 8 and 119868out = 4
iterations does not bring significant improvement on theperformance compared to the case when 119868in = 2 and 119868out =
4 iterations are performed Similarly in the case of LDPCdecoder the performance of 119868in = 20 and 119868out = 4 iterationsis comparable to the performance of 119868in = [3 4 6 7] and119868out = 4 iterations Hence using a large number of iterationsdoes not seem to be efficientwhich proves the results obtainedin the convergence analysis of Section 5
By comparing the algorithms LC-K-Best decoder showsa degradation of less than 02 dB compared to STS-SD ata BER level of 2 times 10
minus5 However it outperforms MMSE-IC and I-VBLAST equalizer by about 02 dB at a BERlevel of 2 times 10
minus5 MMSE-IC and I-VBLAST show almostthe same performance In addition in the case of LDPCdecoder (Figure 8(b)) we notice that increasing the numberof inner iterations for each outer iteration 119868in = [3 4 6 7]
12 Mobile Information Systems
0 05 1 15 2 25 3
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 8
(a) LTE turbo decoder
BER
0 05 1 15 2 25 3
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 20
STS-SD Iin = 5
STS-SD
LC-K-BestI-VBLASTMMSE-IC
(b) LDPC decoder
Figure 8 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
shows slightly better performance than performing an equalnumber of iterations 119868in = 5 for each outer iteration
In the case of high-order modulation higher spectralefficiency can be achieved at a cost of increased symboldetection difficulty Figure 9 shows the BER performance of64-QAM with 119877
119888= 34 We see that LC-K-Best decoder
with a list size of 32 presents the similar performance as STS-SD However I-VLAST equalizer and MMSE-IC equalizerpresent degradation of more than 2 dB at a BER level of 1 times
10minus5 compared to LC-K-Best decoder Therefore LC-K-Best
decoder is more robust in the case of high-ordermodulationsand high coding rates The figure also shows that the BERperformance of LDPC decoder is almost identical to that ofthe turbo decoder
In order to summarize the performance of differentdetectors with different channel decoders we provide the119864119887119873
0values achieving a BER level of 1 times 10
minus5 in Table 4The values given in the parentheses of the table represent theperformance loss compared to STS-SD
Next we evaluate the performance of the iterative receiverin more realistic channel environments Figures 10 11 and 12show the BER performance of the detectors with the channeldecoders on EPA EVA and ETU channels receptivelySimilar behaviors can be observed with LTE turbo decoderand with LDPC decoder
In EPA channel (Figure 10) we see that LC-K-Bestdecoder achieves similar performance as STS-SD in the caseof 64-AQM and presents a degradation less than 02 dB in the
case of 16-QAM Meanwhile MMSE-IC presents significantperformance loss of more than 6 dB in the case of 64-QAMand 119877
119888= 34 With 16-QAM and 119877
119888= 12 the degradation
of MMSE-IC compared to LC-K-Best decoder is about 1 dBat a BER level of 1 times 10
minus4In EVA channel (Figure 11) the performance loss of
MMSE-IC compared to LC-K-Best decoder is reduced toapproximately 5 dB with 64-QAM and 05 dB with 16-QAMLC-K-Best decoder presents a degradation of about 01sim03 dB compared to STS-SD in the case of 16-QAM and 64-QAM
Similarly in ETU channel (Figure 12) MMSE-IC presentsa performance degradation compared to LC-K-Best decoderThis degradation is less than 4 dB in the case of 64-QAM andless than 05 dB in the case of 16-QAM We notice also thatthe LC-K-Best decoder is comparable to STS-SD in the caseof 64-QAM and has a degradation of 02 dB in the case of 16-QAM
Comparing the performance of the iterative receiver indifferent channels it can be seen that iterative receiverspresent the best performance in ETU channel compared toEPA and EVA channels This is due to the high diversity ofETU channel At a BER level of 1 times 10
minus4 the performancegain in ETU channel in the case of LTE turbo decoder is about08 dB 13 dB compared to EVA channel with 16-QAM and64-QAM respectively In the case of LDPC decoder this gainis 04 dB and 1 dB with 16-QAM and 64-QAM respectivelyHowever in EPA channel the performance gain in ETU
Mobile Information Systems 13
3 4 5 6 7 8 9 10 11 12 13
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
EbN0 (dB)
(a) LTE turbo decoder
3 4 5 6 7 8 9 10 11 12
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)
STS-SD Iin = 20
STS-SDLC-K-Best
I-VBLASTMMSE-IC
(b) LDPC decoder
Figure 9 BER performance of a 4 times 4 spatial multiplexing system with 64-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
0values achieving a BER level of 1times 10minus5 for different detectors and channel decoders (turbo LDPC) in 4times 4 spatialmultiplexing
system with 16-QAM 119877119888
= 12 and 64-QAM 119877119888
= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
STS-SD 25 dB 102 dB 27 dB 96 dBLC-119870-Best 26 dB (minus01) 102 dB (00) 28 dB (minus01) 96 dB (00)I-VBLAST 28 dB (minus03) 128 dB (minus26) 29 dB (minus02) 12 dB (minus24)MMSE-IC 28 dB (minus03) 128 dB (minus26) 30 dB (minus03) 12 dB (minus24)lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
channel in the case of turbo decoder or LDPC decoder ismore than 1 dB with 16-QAM and 64-QAM
Table 5 summarizes the 119864119887119873
0values achieving a BER
level of 1times10minus4 of different detectors combined with different
channel decoders andmodulation orders in different channelmodels The values given in the parentheses in the tablerepresent the performance loss compared to STS-SD Asindicated in Table 5 the iterative receivers with turbo decoderand LDPC decoder have comparable performance with acoding rate 119877
119888= 12 (16-QAM) However with 119877
119888= 34
(64-QAM) the receivers with LDPC decoder present slightlya better performance especially in ETU channel (06 dB)
From these results we show that the iterative receiversubstantially improves the performance of coded MIMOsystems either with turbo decoder or with LDPC decoderin Rayleigh channel (Figures 8 and 9) and in more realisticchannels (Figures 10 11 and 12) Moreover we show that
performing a large number of inner iterations does notbring significant improvement In addition we show thatthe LC-K-Best decoder achieves a good performance withdifferent modulations and channel coding schemes Thefigures suggest that the BER performance of the iterativereceiver with turbo decoder is almost comparable to that ofthe LDPC decoder It is therefore meaningful to evaluate thecomputational complexity of the iterative receivers with bothdecoding techniques as it will be discussed in the next section
62 Complexity Evaluation The computational complexityhas significant impact on the latency throughput and powerconsumption of the deviceTherefore the receiver algorithmsshould be optimized to achieve a good trade-off betweenperformance and cost In this part we evaluate the com-putational complexity of the iterative receiver in terms ofbasic operations such as addition subtractionmultiplication
14 Mobile Information Systems
1 3 5 7 9 11 13 15 17 19
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
(a) EPA turbo 119868out = 4 119868in = 2
BER
1 3 5 7 9 11 13 15 17 19
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12 64-QAM
Rc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EPA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 10 BER performance of a 4 times 4 spatial multiplexing with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) on EPA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944 119868out = 4
119868in = [3 4 6 7]
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
STS-SDLC-K-Best
MMSE-IC
16-QAMRc = 12
64-QAMRc = 34
(a) EVA turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EVA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 11 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Best and MMSE-IC) on EVA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944
and 119868out = 4 119868in = [3 4 6 7]
Mobile Information Systems 15
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(a) ETU turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) ETU LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 12 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SDLC-K-Best and MMSE-IC) on ETU channel model (a) LTE turbo decoder 119870
minus4 in LTE channel models for different detectors and channel decoders (turbo LDPC)in 4 times 4 spatial multiplexing system with 16-QAM 119877
119888= 12 and 64-QAM 119877
119888= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
EPASTS-SD 63 dB 140 dB 62 dB 138 dB
LC-119870-Best 65 dB (minus02) 139 dB (+01) 64 dB (minus02) 139 dB (minus01)MMSE-IC 74 dB (minus11) gt20 dB (gtminus6) 80 dB (minus18) gt20 dB (gtminus6)
EVASTS-SD 52 dB 143 dB 53 dB 134 dB
LC-119870-Best 54 dB (minus02) 144 dB (minus01) 56 dB (minus03) 137 dB (minus03)MMSE-IC 58 dB (minus06) 190 dB (minus47) 60 dB (minus07) 185 dB (minus51)
ETUSTS-SD 44 dB 130 dB 49 dB 124 dB
LC-119870-Best 46 dB (minus02) 130 dB (00) 49 dB (00) 124 dB (00)MMSE-IC 49 dB (minus05) 175 dB (minus35) 51 dB (minus02) 16 dB (minus36)
lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
division square root extraction maximization and look-up table check (which are denoted by ADD SUB MULDIV SQRT Max and LUT resp) To this end the completecomparison of the iterative receivers with both channeldecoders (turbo LDPC) and with several modulations andcoding rates is carried out
621 Complexity of Iterative Receiver The complexity ofiterative receiver depends on theMIMOdetector the channeldecoder and the number of innerouter iterationsThis com-plexity can be expressed by
sdot [119862det1 + (119868out minus 1) sdot 119862deti] (34)
where 119862det1 denotes the complexity of the first iterationof MIMO detection algorithm per symbol vector withouttaking into consideration the a priori information 119862detidenotes the complexity per iteration per symbol vector takinginto consideration the a priori information 119862dec denotesthe complexity of the channel decoder per iteration perinformation bit 119873bit is the number of information bit at theinput of the encoder 119873symb is the number of symbol vectors119873symb and 119873bit are linked by the following relation
119873symb =119873bit
119876119877119888119873119905
= 120572119873bit with 120572 =1
119876119877119888119873119905
(35)
where 119876 is the number of bits in the constellation symbol 119877119888
is the coding rate and119873119905is the number of transmit antennas
16 Mobile Information Systems
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
error rate (BER) with respect to signal-to-noise ratio (SNR)per bit 119864
119887119873
0
119864119887
1198730
=119864119904
1198730
+ 10 sdot log10
1
119877119888119876119873
119905
[dB] (33)
In our previous study [24] the performance of MIMOdetectors with LTE turbo decoder is evaluated in a Rayleighchannel with various outer and inner iterations It has beenshown that the performance is improved by about 15 dB at aBER level of 1 times 10
minus5 with 4 outer iterations It has been alsoshown that no significant improvement can be achieved after4 outer iterations this improvement is less than 02 dB with 8outer iterations
Similarly to turbo decoder we fix the number of inneriterations inside LDPC decoder to 20 while varying thenumber of outer iterations Figure 7 shows BER performanceof MIMO detectors with LDPC decoder in Rayleigh channelwith 119868in = 20 iterations and 119868out = 1 2 4 or 8 iterationsSTS-SD is used with a LLR clipping level of 8 which givesclose toMAPperformancewith considerable reduction in thecomplexity For LC-K-Best decoder a LLR clipping level of 3is used in the case of missing counter hypothesis We show
0 1 2 3 4 5
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
Iout = 4
Iout = 8
Iout = 2
Iout = 1
EbN0 (dB)minus1
Figure 7 BER performance of a 4 times 4 spatial multiplexing systemwith 16-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) in a Rayleigh fading channel 119868out = 1 2 4 8 119868in =
20 LDPC decoder 119873119887
= 1944 and 119877119888
= 12
that performance improvement of 15 dB is observed with 4outer iterations For 119868out = 8 iterations the improvementis less than 02 dB Therefore 119868out = 4 iterations will beconsidered in the sequel
Figure 8 shows the BER performance of 4 times 4 16-QAMsystem in a Rayleigh fading channel with 119868out = 4 and 119868in = 2
in the case of turbo decoder and 119868in = [3 4 6 7] in the caseof LDPC decoder The notation 119868in = [3 4 6 7] denotes that3 inner iterations are performed in the 1st outer iteration 4inner iterations in the 2nd outer iteration and so on Theperformances of STS-SD with 119868in = 8 and 119868in = 20 foreach outer iteration in the case of turbo decoder and LDPCdecoder are also plotted as a reference In the case of turbodecoder we show that performing 119868in = 8 and 119868out = 4
iterations does not bring significant improvement on theperformance compared to the case when 119868in = 2 and 119868out =
4 iterations are performed Similarly in the case of LDPCdecoder the performance of 119868in = 20 and 119868out = 4 iterationsis comparable to the performance of 119868in = [3 4 6 7] and119868out = 4 iterations Hence using a large number of iterationsdoes not seem to be efficientwhich proves the results obtainedin the convergence analysis of Section 5
By comparing the algorithms LC-K-Best decoder showsa degradation of less than 02 dB compared to STS-SD ata BER level of 2 times 10
minus5 However it outperforms MMSE-IC and I-VBLAST equalizer by about 02 dB at a BERlevel of 2 times 10
minus5 MMSE-IC and I-VBLAST show almostthe same performance In addition in the case of LDPCdecoder (Figure 8(b)) we notice that increasing the numberof inner iterations for each outer iteration 119868in = [3 4 6 7]
12 Mobile Information Systems
0 05 1 15 2 25 3
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 8
(a) LTE turbo decoder
BER
0 05 1 15 2 25 3
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1 minus05
STS-SD Iin = 20
STS-SD Iin = 5
STS-SD
LC-K-BestI-VBLASTMMSE-IC
(b) LDPC decoder
Figure 8 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
shows slightly better performance than performing an equalnumber of iterations 119868in = 5 for each outer iteration
In the case of high-order modulation higher spectralefficiency can be achieved at a cost of increased symboldetection difficulty Figure 9 shows the BER performance of64-QAM with 119877
119888= 34 We see that LC-K-Best decoder
with a list size of 32 presents the similar performance as STS-SD However I-VLAST equalizer and MMSE-IC equalizerpresent degradation of more than 2 dB at a BER level of 1 times
10minus5 compared to LC-K-Best decoder Therefore LC-K-Best
decoder is more robust in the case of high-ordermodulationsand high coding rates The figure also shows that the BERperformance of LDPC decoder is almost identical to that ofthe turbo decoder
In order to summarize the performance of differentdetectors with different channel decoders we provide the119864119887119873
0values achieving a BER level of 1 times 10
minus5 in Table 4The values given in the parentheses of the table represent theperformance loss compared to STS-SD
Next we evaluate the performance of the iterative receiverin more realistic channel environments Figures 10 11 and 12show the BER performance of the detectors with the channeldecoders on EPA EVA and ETU channels receptivelySimilar behaviors can be observed with LTE turbo decoderand with LDPC decoder
In EPA channel (Figure 10) we see that LC-K-Bestdecoder achieves similar performance as STS-SD in the caseof 64-AQM and presents a degradation less than 02 dB in the
case of 16-QAM Meanwhile MMSE-IC presents significantperformance loss of more than 6 dB in the case of 64-QAMand 119877
119888= 34 With 16-QAM and 119877
119888= 12 the degradation
of MMSE-IC compared to LC-K-Best decoder is about 1 dBat a BER level of 1 times 10
minus4In EVA channel (Figure 11) the performance loss of
MMSE-IC compared to LC-K-Best decoder is reduced toapproximately 5 dB with 64-QAM and 05 dB with 16-QAMLC-K-Best decoder presents a degradation of about 01sim03 dB compared to STS-SD in the case of 16-QAM and 64-QAM
Similarly in ETU channel (Figure 12) MMSE-IC presentsa performance degradation compared to LC-K-Best decoderThis degradation is less than 4 dB in the case of 64-QAM andless than 05 dB in the case of 16-QAM We notice also thatthe LC-K-Best decoder is comparable to STS-SD in the caseof 64-QAM and has a degradation of 02 dB in the case of 16-QAM
Comparing the performance of the iterative receiver indifferent channels it can be seen that iterative receiverspresent the best performance in ETU channel compared toEPA and EVA channels This is due to the high diversity ofETU channel At a BER level of 1 times 10
minus4 the performancegain in ETU channel in the case of LTE turbo decoder is about08 dB 13 dB compared to EVA channel with 16-QAM and64-QAM respectively In the case of LDPC decoder this gainis 04 dB and 1 dB with 16-QAM and 64-QAM respectivelyHowever in EPA channel the performance gain in ETU
Mobile Information Systems 13
3 4 5 6 7 8 9 10 11 12 13
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
EbN0 (dB)
(a) LTE turbo decoder
3 4 5 6 7 8 9 10 11 12
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)
STS-SD Iin = 20
STS-SDLC-K-Best
I-VBLASTMMSE-IC
(b) LDPC decoder
Figure 9 BER performance of a 4 times 4 spatial multiplexing system with 64-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
0values achieving a BER level of 1times 10minus5 for different detectors and channel decoders (turbo LDPC) in 4times 4 spatialmultiplexing
system with 16-QAM 119877119888
= 12 and 64-QAM 119877119888
= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
STS-SD 25 dB 102 dB 27 dB 96 dBLC-119870-Best 26 dB (minus01) 102 dB (00) 28 dB (minus01) 96 dB (00)I-VBLAST 28 dB (minus03) 128 dB (minus26) 29 dB (minus02) 12 dB (minus24)MMSE-IC 28 dB (minus03) 128 dB (minus26) 30 dB (minus03) 12 dB (minus24)lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
channel in the case of turbo decoder or LDPC decoder ismore than 1 dB with 16-QAM and 64-QAM
Table 5 summarizes the 119864119887119873
0values achieving a BER
level of 1times10minus4 of different detectors combined with different
channel decoders andmodulation orders in different channelmodels The values given in the parentheses in the tablerepresent the performance loss compared to STS-SD Asindicated in Table 5 the iterative receivers with turbo decoderand LDPC decoder have comparable performance with acoding rate 119877
119888= 12 (16-QAM) However with 119877
119888= 34
(64-QAM) the receivers with LDPC decoder present slightlya better performance especially in ETU channel (06 dB)
From these results we show that the iterative receiversubstantially improves the performance of coded MIMOsystems either with turbo decoder or with LDPC decoderin Rayleigh channel (Figures 8 and 9) and in more realisticchannels (Figures 10 11 and 12) Moreover we show that
performing a large number of inner iterations does notbring significant improvement In addition we show thatthe LC-K-Best decoder achieves a good performance withdifferent modulations and channel coding schemes Thefigures suggest that the BER performance of the iterativereceiver with turbo decoder is almost comparable to that ofthe LDPC decoder It is therefore meaningful to evaluate thecomputational complexity of the iterative receivers with bothdecoding techniques as it will be discussed in the next section
62 Complexity Evaluation The computational complexityhas significant impact on the latency throughput and powerconsumption of the deviceTherefore the receiver algorithmsshould be optimized to achieve a good trade-off betweenperformance and cost In this part we evaluate the com-putational complexity of the iterative receiver in terms ofbasic operations such as addition subtractionmultiplication
14 Mobile Information Systems
1 3 5 7 9 11 13 15 17 19
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
(a) EPA turbo 119868out = 4 119868in = 2
BER
1 3 5 7 9 11 13 15 17 19
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12 64-QAM
Rc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EPA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 10 BER performance of a 4 times 4 spatial multiplexing with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) on EPA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944 119868out = 4
119868in = [3 4 6 7]
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
STS-SDLC-K-Best
MMSE-IC
16-QAMRc = 12
64-QAMRc = 34
(a) EVA turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EVA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 11 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Best and MMSE-IC) on EVA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944
and 119868out = 4 119868in = [3 4 6 7]
Mobile Information Systems 15
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(a) ETU turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) ETU LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 12 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SDLC-K-Best and MMSE-IC) on ETU channel model (a) LTE turbo decoder 119870
minus4 in LTE channel models for different detectors and channel decoders (turbo LDPC)in 4 times 4 spatial multiplexing system with 16-QAM 119877
119888= 12 and 64-QAM 119877
119888= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
EPASTS-SD 63 dB 140 dB 62 dB 138 dB
LC-119870-Best 65 dB (minus02) 139 dB (+01) 64 dB (minus02) 139 dB (minus01)MMSE-IC 74 dB (minus11) gt20 dB (gtminus6) 80 dB (minus18) gt20 dB (gtminus6)
EVASTS-SD 52 dB 143 dB 53 dB 134 dB
LC-119870-Best 54 dB (minus02) 144 dB (minus01) 56 dB (minus03) 137 dB (minus03)MMSE-IC 58 dB (minus06) 190 dB (minus47) 60 dB (minus07) 185 dB (minus51)
ETUSTS-SD 44 dB 130 dB 49 dB 124 dB
LC-119870-Best 46 dB (minus02) 130 dB (00) 49 dB (00) 124 dB (00)MMSE-IC 49 dB (minus05) 175 dB (minus35) 51 dB (minus02) 16 dB (minus36)
lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
division square root extraction maximization and look-up table check (which are denoted by ADD SUB MULDIV SQRT Max and LUT resp) To this end the completecomparison of the iterative receivers with both channeldecoders (turbo LDPC) and with several modulations andcoding rates is carried out
621 Complexity of Iterative Receiver The complexity ofiterative receiver depends on theMIMOdetector the channeldecoder and the number of innerouter iterationsThis com-plexity can be expressed by
sdot [119862det1 + (119868out minus 1) sdot 119862deti] (34)
where 119862det1 denotes the complexity of the first iterationof MIMO detection algorithm per symbol vector withouttaking into consideration the a priori information 119862detidenotes the complexity per iteration per symbol vector takinginto consideration the a priori information 119862dec denotesthe complexity of the channel decoder per iteration perinformation bit 119873bit is the number of information bit at theinput of the encoder 119873symb is the number of symbol vectors119873symb and 119873bit are linked by the following relation
119873symb =119873bit
119876119877119888119873119905
= 120572119873bit with 120572 =1
119876119877119888119873119905
(35)
where 119876 is the number of bits in the constellation symbol 119877119888
is the coding rate and119873119905is the number of transmit antennas
16 Mobile Information Systems
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
Figure 8 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
shows slightly better performance than performing an equalnumber of iterations 119868in = 5 for each outer iteration
In the case of high-order modulation higher spectralefficiency can be achieved at a cost of increased symboldetection difficulty Figure 9 shows the BER performance of64-QAM with 119877
119888= 34 We see that LC-K-Best decoder
with a list size of 32 presents the similar performance as STS-SD However I-VLAST equalizer and MMSE-IC equalizerpresent degradation of more than 2 dB at a BER level of 1 times
10minus5 compared to LC-K-Best decoder Therefore LC-K-Best
decoder is more robust in the case of high-ordermodulationsand high coding rates The figure also shows that the BERperformance of LDPC decoder is almost identical to that ofthe turbo decoder
In order to summarize the performance of differentdetectors with different channel decoders we provide the119864119887119873
0values achieving a BER level of 1 times 10
minus5 in Table 4The values given in the parentheses of the table represent theperformance loss compared to STS-SD
Next we evaluate the performance of the iterative receiverin more realistic channel environments Figures 10 11 and 12show the BER performance of the detectors with the channeldecoders on EPA EVA and ETU channels receptivelySimilar behaviors can be observed with LTE turbo decoderand with LDPC decoder
In EPA channel (Figure 10) we see that LC-K-Bestdecoder achieves similar performance as STS-SD in the caseof 64-AQM and presents a degradation less than 02 dB in the
case of 16-QAM Meanwhile MMSE-IC presents significantperformance loss of more than 6 dB in the case of 64-QAMand 119877
119888= 34 With 16-QAM and 119877
119888= 12 the degradation
of MMSE-IC compared to LC-K-Best decoder is about 1 dBat a BER level of 1 times 10
minus4In EVA channel (Figure 11) the performance loss of
MMSE-IC compared to LC-K-Best decoder is reduced toapproximately 5 dB with 64-QAM and 05 dB with 16-QAMLC-K-Best decoder presents a degradation of about 01sim03 dB compared to STS-SD in the case of 16-QAM and 64-QAM
Similarly in ETU channel (Figure 12) MMSE-IC presentsa performance degradation compared to LC-K-Best decoderThis degradation is less than 4 dB in the case of 64-QAM andless than 05 dB in the case of 16-QAM We notice also thatthe LC-K-Best decoder is comparable to STS-SD in the caseof 64-QAM and has a degradation of 02 dB in the case of 16-QAM
Comparing the performance of the iterative receiver indifferent channels it can be seen that iterative receiverspresent the best performance in ETU channel compared toEPA and EVA channels This is due to the high diversity ofETU channel At a BER level of 1 times 10
minus4 the performancegain in ETU channel in the case of LTE turbo decoder is about08 dB 13 dB compared to EVA channel with 16-QAM and64-QAM respectively In the case of LDPC decoder this gainis 04 dB and 1 dB with 16-QAM and 64-QAM respectivelyHowever in EPA channel the performance gain in ETU
Mobile Information Systems 13
3 4 5 6 7 8 9 10 11 12 13
BER
STS-SDLC-K-Best
I-VBLASTMMSE-IC
100
10minus1
10minus2
10minus3
10minus4
EbN0 (dB)
(a) LTE turbo decoder
3 4 5 6 7 8 9 10 11 12
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)
STS-SD Iin = 20
STS-SDLC-K-Best
I-VBLASTMMSE-IC
(b) LDPC decoder
Figure 9 BER performance of a 4 times 4 spatial multiplexing system with 64-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
0values achieving a BER level of 1times 10minus5 for different detectors and channel decoders (turbo LDPC) in 4times 4 spatialmultiplexing
system with 16-QAM 119877119888
= 12 and 64-QAM 119877119888
= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
STS-SD 25 dB 102 dB 27 dB 96 dBLC-119870-Best 26 dB (minus01) 102 dB (00) 28 dB (minus01) 96 dB (00)I-VBLAST 28 dB (minus03) 128 dB (minus26) 29 dB (minus02) 12 dB (minus24)MMSE-IC 28 dB (minus03) 128 dB (minus26) 30 dB (minus03) 12 dB (minus24)lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
channel in the case of turbo decoder or LDPC decoder ismore than 1 dB with 16-QAM and 64-QAM
Table 5 summarizes the 119864119887119873
0values achieving a BER
level of 1times10minus4 of different detectors combined with different
channel decoders andmodulation orders in different channelmodels The values given in the parentheses in the tablerepresent the performance loss compared to STS-SD Asindicated in Table 5 the iterative receivers with turbo decoderand LDPC decoder have comparable performance with acoding rate 119877
119888= 12 (16-QAM) However with 119877
119888= 34
(64-QAM) the receivers with LDPC decoder present slightlya better performance especially in ETU channel (06 dB)
From these results we show that the iterative receiversubstantially improves the performance of coded MIMOsystems either with turbo decoder or with LDPC decoderin Rayleigh channel (Figures 8 and 9) and in more realisticchannels (Figures 10 11 and 12) Moreover we show that
performing a large number of inner iterations does notbring significant improvement In addition we show thatthe LC-K-Best decoder achieves a good performance withdifferent modulations and channel coding schemes Thefigures suggest that the BER performance of the iterativereceiver with turbo decoder is almost comparable to that ofthe LDPC decoder It is therefore meaningful to evaluate thecomputational complexity of the iterative receivers with bothdecoding techniques as it will be discussed in the next section
62 Complexity Evaluation The computational complexityhas significant impact on the latency throughput and powerconsumption of the deviceTherefore the receiver algorithmsshould be optimized to achieve a good trade-off betweenperformance and cost In this part we evaluate the com-putational complexity of the iterative receiver in terms ofbasic operations such as addition subtractionmultiplication
14 Mobile Information Systems
1 3 5 7 9 11 13 15 17 19
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
(a) EPA turbo 119868out = 4 119868in = 2
BER
1 3 5 7 9 11 13 15 17 19
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12 64-QAM
Rc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EPA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 10 BER performance of a 4 times 4 spatial multiplexing with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) on EPA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944 119868out = 4
119868in = [3 4 6 7]
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
STS-SDLC-K-Best
MMSE-IC
16-QAMRc = 12
64-QAMRc = 34
(a) EVA turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EVA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 11 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Best and MMSE-IC) on EVA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944
and 119868out = 4 119868in = [3 4 6 7]
Mobile Information Systems 15
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(a) ETU turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) ETU LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 12 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SDLC-K-Best and MMSE-IC) on ETU channel model (a) LTE turbo decoder 119870
minus4 in LTE channel models for different detectors and channel decoders (turbo LDPC)in 4 times 4 spatial multiplexing system with 16-QAM 119877
119888= 12 and 64-QAM 119877
119888= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
EPASTS-SD 63 dB 140 dB 62 dB 138 dB
LC-119870-Best 65 dB (minus02) 139 dB (+01) 64 dB (minus02) 139 dB (minus01)MMSE-IC 74 dB (minus11) gt20 dB (gtminus6) 80 dB (minus18) gt20 dB (gtminus6)
EVASTS-SD 52 dB 143 dB 53 dB 134 dB
LC-119870-Best 54 dB (minus02) 144 dB (minus01) 56 dB (minus03) 137 dB (minus03)MMSE-IC 58 dB (minus06) 190 dB (minus47) 60 dB (minus07) 185 dB (minus51)
ETUSTS-SD 44 dB 130 dB 49 dB 124 dB
LC-119870-Best 46 dB (minus02) 130 dB (00) 49 dB (00) 124 dB (00)MMSE-IC 49 dB (minus05) 175 dB (minus35) 51 dB (minus02) 16 dB (minus36)
lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
division square root extraction maximization and look-up table check (which are denoted by ADD SUB MULDIV SQRT Max and LUT resp) To this end the completecomparison of the iterative receivers with both channeldecoders (turbo LDPC) and with several modulations andcoding rates is carried out
621 Complexity of Iterative Receiver The complexity ofiterative receiver depends on theMIMOdetector the channeldecoder and the number of innerouter iterationsThis com-plexity can be expressed by
sdot [119862det1 + (119868out minus 1) sdot 119862deti] (34)
where 119862det1 denotes the complexity of the first iterationof MIMO detection algorithm per symbol vector withouttaking into consideration the a priori information 119862detidenotes the complexity per iteration per symbol vector takinginto consideration the a priori information 119862dec denotesthe complexity of the channel decoder per iteration perinformation bit 119873bit is the number of information bit at theinput of the encoder 119873symb is the number of symbol vectors119873symb and 119873bit are linked by the following relation
119873symb =119873bit
119876119877119888119873119905
= 120572119873bit with 120572 =1
119876119877119888119873119905
(35)
where 119876 is the number of bits in the constellation symbol 119877119888
is the coding rate and119873119905is the number of transmit antennas
16 Mobile Information Systems
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
Figure 9 BER performance of a 4 times 4 spatial multiplexing system with 64-QAM using several MIMO detectors (STS-SD LC-K-Best I-VBLAST and MMSE-IC) in a Rayleigh fading channel (a) LTE turbo decoder 119870
0values achieving a BER level of 1times 10minus5 for different detectors and channel decoders (turbo LDPC) in 4times 4 spatialmultiplexing
system with 16-QAM 119877119888
= 12 and 64-QAM 119877119888
= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
STS-SD 25 dB 102 dB 27 dB 96 dBLC-119870-Best 26 dB (minus01) 102 dB (00) 28 dB (minus01) 96 dB (00)I-VBLAST 28 dB (minus03) 128 dB (minus26) 29 dB (minus02) 12 dB (minus24)MMSE-IC 28 dB (minus03) 128 dB (minus26) 30 dB (minus03) 12 dB (minus24)lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
channel in the case of turbo decoder or LDPC decoder ismore than 1 dB with 16-QAM and 64-QAM
Table 5 summarizes the 119864119887119873
0values achieving a BER
level of 1times10minus4 of different detectors combined with different
channel decoders andmodulation orders in different channelmodels The values given in the parentheses in the tablerepresent the performance loss compared to STS-SD Asindicated in Table 5 the iterative receivers with turbo decoderand LDPC decoder have comparable performance with acoding rate 119877
119888= 12 (16-QAM) However with 119877
119888= 34
(64-QAM) the receivers with LDPC decoder present slightlya better performance especially in ETU channel (06 dB)
From these results we show that the iterative receiversubstantially improves the performance of coded MIMOsystems either with turbo decoder or with LDPC decoderin Rayleigh channel (Figures 8 and 9) and in more realisticchannels (Figures 10 11 and 12) Moreover we show that
performing a large number of inner iterations does notbring significant improvement In addition we show thatthe LC-K-Best decoder achieves a good performance withdifferent modulations and channel coding schemes Thefigures suggest that the BER performance of the iterativereceiver with turbo decoder is almost comparable to that ofthe LDPC decoder It is therefore meaningful to evaluate thecomputational complexity of the iterative receivers with bothdecoding techniques as it will be discussed in the next section
62 Complexity Evaluation The computational complexityhas significant impact on the latency throughput and powerconsumption of the deviceTherefore the receiver algorithmsshould be optimized to achieve a good trade-off betweenperformance and cost In this part we evaluate the com-putational complexity of the iterative receiver in terms ofbasic operations such as addition subtractionmultiplication
14 Mobile Information Systems
1 3 5 7 9 11 13 15 17 19
BER
STS-SDLC-K-Best
MMSE-IC
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
(a) EPA turbo 119868out = 4 119868in = 2
BER
1 3 5 7 9 11 13 15 17 19
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12 64-QAM
Rc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EPA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 10 BER performance of a 4 times 4 spatial multiplexing with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) on EPA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944 119868out = 4
119868in = [3 4 6 7]
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
STS-SDLC-K-Best
MMSE-IC
16-QAMRc = 12
64-QAMRc = 34
(a) EVA turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EVA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 11 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Best and MMSE-IC) on EVA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944
and 119868out = 4 119868in = [3 4 6 7]
Mobile Information Systems 15
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(a) ETU turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) ETU LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 12 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SDLC-K-Best and MMSE-IC) on ETU channel model (a) LTE turbo decoder 119870
minus4 in LTE channel models for different detectors and channel decoders (turbo LDPC)in 4 times 4 spatial multiplexing system with 16-QAM 119877
119888= 12 and 64-QAM 119877
119888= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
EPASTS-SD 63 dB 140 dB 62 dB 138 dB
LC-119870-Best 65 dB (minus02) 139 dB (+01) 64 dB (minus02) 139 dB (minus01)MMSE-IC 74 dB (minus11) gt20 dB (gtminus6) 80 dB (minus18) gt20 dB (gtminus6)
EVASTS-SD 52 dB 143 dB 53 dB 134 dB
LC-119870-Best 54 dB (minus02) 144 dB (minus01) 56 dB (minus03) 137 dB (minus03)MMSE-IC 58 dB (minus06) 190 dB (minus47) 60 dB (minus07) 185 dB (minus51)
ETUSTS-SD 44 dB 130 dB 49 dB 124 dB
LC-119870-Best 46 dB (minus02) 130 dB (00) 49 dB (00) 124 dB (00)MMSE-IC 49 dB (minus05) 175 dB (minus35) 51 dB (minus02) 16 dB (minus36)
lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
division square root extraction maximization and look-up table check (which are denoted by ADD SUB MULDIV SQRT Max and LUT resp) To this end the completecomparison of the iterative receivers with both channeldecoders (turbo LDPC) and with several modulations andcoding rates is carried out
621 Complexity of Iterative Receiver The complexity ofiterative receiver depends on theMIMOdetector the channeldecoder and the number of innerouter iterationsThis com-plexity can be expressed by
sdot [119862det1 + (119868out minus 1) sdot 119862deti] (34)
where 119862det1 denotes the complexity of the first iterationof MIMO detection algorithm per symbol vector withouttaking into consideration the a priori information 119862detidenotes the complexity per iteration per symbol vector takinginto consideration the a priori information 119862dec denotesthe complexity of the channel decoder per iteration perinformation bit 119873bit is the number of information bit at theinput of the encoder 119873symb is the number of symbol vectors119873symb and 119873bit are linked by the following relation
119873symb =119873bit
119876119877119888119873119905
= 120572119873bit with 120572 =1
119876119877119888119873119905
(35)
where 119876 is the number of bits in the constellation symbol 119877119888
is the coding rate and119873119905is the number of transmit antennas
16 Mobile Information Systems
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
Figure 10 BER performance of a 4 times 4 spatial multiplexing with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Bestand MMSE-IC) on EPA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944 119868out = 4
119868in = [3 4 6 7]
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
STS-SDLC-K-Best
MMSE-IC
16-QAMRc = 12
64-QAMRc = 34
(a) EVA turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) EVA LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 11 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SD LC-K-Best and MMSE-IC) on EVA channel model (a) LTE turbo decoder 119870
119887= 1024 and 119868out = 4 119868in = 2 and (b) LDPC decoder 119873
119887= 1944
and 119868out = 4 119868in = [3 4 6 7]
Mobile Information Systems 15
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(a) ETU turbo 119868out = 4 119868in = 2
1 3 5 7 9 11 13 15 17 19
BER
100
10minus1
10minus2
10minus3
10minus4
10minus5
EbN0 (dB)minus1
16-QAMRc = 12
64-QAMRc = 34
STS-SDLC-K-Best
MMSE-IC
(b) ETU LDPC 119868out = 4 119868in = [3 4 6 7]
Figure 12 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SDLC-K-Best and MMSE-IC) on ETU channel model (a) LTE turbo decoder 119870
minus4 in LTE channel models for different detectors and channel decoders (turbo LDPC)in 4 times 4 spatial multiplexing system with 16-QAM 119877
119888= 12 and 64-QAM 119877
119888= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
EPASTS-SD 63 dB 140 dB 62 dB 138 dB
LC-119870-Best 65 dB (minus02) 139 dB (+01) 64 dB (minus02) 139 dB (minus01)MMSE-IC 74 dB (minus11) gt20 dB (gtminus6) 80 dB (minus18) gt20 dB (gtminus6)
EVASTS-SD 52 dB 143 dB 53 dB 134 dB
LC-119870-Best 54 dB (minus02) 144 dB (minus01) 56 dB (minus03) 137 dB (minus03)MMSE-IC 58 dB (minus06) 190 dB (minus47) 60 dB (minus07) 185 dB (minus51)
ETUSTS-SD 44 dB 130 dB 49 dB 124 dB
LC-119870-Best 46 dB (minus02) 130 dB (00) 49 dB (00) 124 dB (00)MMSE-IC 49 dB (minus05) 175 dB (minus35) 51 dB (minus02) 16 dB (minus36)
lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
division square root extraction maximization and look-up table check (which are denoted by ADD SUB MULDIV SQRT Max and LUT resp) To this end the completecomparison of the iterative receivers with both channeldecoders (turbo LDPC) and with several modulations andcoding rates is carried out
621 Complexity of Iterative Receiver The complexity ofiterative receiver depends on theMIMOdetector the channeldecoder and the number of innerouter iterationsThis com-plexity can be expressed by
sdot [119862det1 + (119868out minus 1) sdot 119862deti] (34)
where 119862det1 denotes the complexity of the first iterationof MIMO detection algorithm per symbol vector withouttaking into consideration the a priori information 119862detidenotes the complexity per iteration per symbol vector takinginto consideration the a priori information 119862dec denotesthe complexity of the channel decoder per iteration perinformation bit 119873bit is the number of information bit at theinput of the encoder 119873symb is the number of symbol vectors119873symb and 119873bit are linked by the following relation
119873symb =119873bit
119876119877119888119873119905
= 120572119873bit with 120572 =1
119876119877119888119873119905
(35)
where 119876 is the number of bits in the constellation symbol 119877119888
is the coding rate and119873119905is the number of transmit antennas
16 Mobile Information Systems
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
Figure 12 BER performance of a 4 times 4 spatial multiplexing system with 16-QAM and 64-QAM using several MIMO detectors (STS-SDLC-K-Best and MMSE-IC) on ETU channel model (a) LTE turbo decoder 119870
minus4 in LTE channel models for different detectors and channel decoders (turbo LDPC)in 4 times 4 spatial multiplexing system with 16-QAM 119877
119888= 12 and 64-QAM 119877
119888= 34lowast
Turbo code LDPC code16-QAM 64-QAM 16-QAM 64-QAM
EPASTS-SD 63 dB 140 dB 62 dB 138 dB
LC-119870-Best 65 dB (minus02) 139 dB (+01) 64 dB (minus02) 139 dB (minus01)MMSE-IC 74 dB (minus11) gt20 dB (gtminus6) 80 dB (minus18) gt20 dB (gtminus6)
EVASTS-SD 52 dB 143 dB 53 dB 134 dB
LC-119870-Best 54 dB (minus02) 144 dB (minus01) 56 dB (minus03) 137 dB (minus03)MMSE-IC 58 dB (minus06) 190 dB (minus47) 60 dB (minus07) 185 dB (minus51)
ETUSTS-SD 44 dB 130 dB 49 dB 124 dB
LC-119870-Best 46 dB (minus02) 130 dB (00) 49 dB (00) 124 dB (00)MMSE-IC 49 dB (minus05) 175 dB (minus35) 51 dB (minus02) 16 dB (minus36)
lowastThe number in the parenthesis corresponds to the performance loss in dB compared to STS-SD
division square root extraction maximization and look-up table check (which are denoted by ADD SUB MULDIV SQRT Max and LUT resp) To this end the completecomparison of the iterative receivers with both channeldecoders (turbo LDPC) and with several modulations andcoding rates is carried out
621 Complexity of Iterative Receiver The complexity ofiterative receiver depends on theMIMOdetector the channeldecoder and the number of innerouter iterationsThis com-plexity can be expressed by
sdot [119862det1 + (119868out minus 1) sdot 119862deti] (34)
where 119862det1 denotes the complexity of the first iterationof MIMO detection algorithm per symbol vector withouttaking into consideration the a priori information 119862detidenotes the complexity per iteration per symbol vector takinginto consideration the a priori information 119862dec denotesthe complexity of the channel decoder per iteration perinformation bit 119873bit is the number of information bit at theinput of the encoder 119873symb is the number of symbol vectors119873symb and 119873bit are linked by the following relation
119873symb =119873bit
119876119877119888119873119905
= 120572119873bit with 120572 =1
119876119877119888119873119905
(35)
where 119876 is the number of bits in the constellation symbol 119877119888
is the coding rate and119873119905is the number of transmit antennas
16 Mobile Information Systems
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
Table 6 Complexity of turbo decoder per information bit periteration
ADDSUB Max (2-input) LUTGamma 3 mdash mdashBeta alpha 42
1198982119898
2119898
LLR 62119898
+ 1 4(2119898
minus 1) 4(2119898
minus 1)
Turbo decoder 282119898
+ 8 122119898
minus 8 122119898
minus 8
LTE turbo decoder 2119898
= 8 232 88 88
622 Channel Decoder Complexity The complexity of turbodecoder depends on the SISO decoder algorithms and thenumber of iterations Herein max-log-MAP algorithm witha correction factor is used [30] The complexity of max-log-MAP decoder corresponds to three principal computationsbranch metrics recursive state metrics and LLR of the bits
Table 6 summarizes the total number of operations perinformation bit per iteration for the LTE turbo decoder with2119898
= 8 states and 119899 = 2 output bits where 119898 is thememory length of the component encoder Therefore theoverall complexity of the turbo decoder can be obtained bymultiplying the information block length 119870
119887and the number
of iterations 119868in sdot 119868outThe complexity of LDPC decoder depends on the
scheduling used to exchange the messages between checknode (CN) and variable node (VN) There are two distinctschedules of belief propagation flooding schedule and lay-ered schedule In the flooding schedule the messages arepassed back and forth along all the edges This scheduleincreases the complexity especially with long block lengthA layered schedule is therefore proposed where only a smallnumber of check nodes and variable nodes are updated persubiteration [53] The messages generated in a subiterationare immediately used in subsequent subiterations of currentiterationThis leads to a faster convergence of LDPCdecodingand a reduction of the required memory size
The computational complexity of the layered LDPCdecoder can be expressed in the function of degree ofconnectivity as summarized in Table 7 119889V119894 and 119889
119888119895denote the
degree of connectivity of the variable node 119894 and the checknode 119895 respectively 119889
119888and 119889V denote the average row weight
and the average column weight of LDPC code respectively
623 Iterative MIMO Detection Complexity The computa-tional complexity of MIMO detection depends on the detec-tion algorithm In the case of tree-search-based algorithmsthe commonly used approach to measure the complexityis to count the number of visited nodes in the tree-searchprocess [54ndash56] However in the case of the interference-cancellation-based equalizers the complexity is evaluated interms of real or complex operations required to computefilter coefficients For a fair comparison the complexity isestimated based on basic operations (ADD SUB MUL DIVSQRT Max and LUT) in this work
The complexity of tree-search-based algorithms can bedivided into two steps the preprocessing and the tree-search process The complexity of interference-cancellation-based equalizer algorithms is dominated by the computation
of the filter coefficients and the matrix inversion Severalmethods for matrix inversion namely Cholesky decompo-sition and QR decomposition have been widely studied inthe literature Herein QR decomposition based on Gram-Schmidt method is used to compute the matrix inversionHowever more efficient method for QR decomposition maybe considered to optimize the cost of computational complex-ity in hardware implementation like Givens rotations (GR)that can be effectively done by coordinate rotation digitalcomputer (CORDIC) scheme
In the case of STS-SD it is very difficult to find ananalytical expression of the complexity due to the sequentialnature of the tree search and the channel statisticsThereforeMonte Carlo simulations were used to measure the averagenumber of operations of STS-SD over all SNR range
The complexity of the interference-cancellation-basedequalization comprises the complexity of soft mapping andsoft demapping In the case of STS-SD and LC-K-Bestdecoder the computational complexity includes the com-plexity of SQR decomposition for the first iteration and thecomplexity of LLR computation The SQR decompositionis based on Gram-Schmidt method which requires manyADD MUL DIV and SQRT operations It is importantto note that in LC-K-Best decoder there is a number ofcomparisons to choose the best candidates that are not takeninto consideration in the complexity comparisons
Figure 13 summarizes the complexity of different detec-tion algorithms in terms of number of operations in thecase of 4 times 4 spatial multiplexing system using 16-QAM forthe 1st and 119894th iteration The MAP algorithm presents thehighest complexity (47 times 10
6 MUL 46 times 106 ADD) It is
not represented in the graph but it is used as a referenceto view the reduction in the complexity of other algorithmscompared to the optimal detector The average number ofarithmetic operations of STS-SD is 90 lower than the MAPalgorithm However it still has a larger complexity than otheralgorithms The complexity of LC-K-Best is approximately30 higher than that of the MMSE equalizer and 50 lowerthan that of I-VBLAST I-VBLAST requires more complexitydue to the matrix inversion for each detected symbol Atthe 119894th iteration LC-MMSE-IC algorithm proposed in [9]has slightly lower complexity than the LC-K-Best decoder interms ofMUL (7) and ADD (19) with additional DIV andSQRT operations required for the matrix inversion
Figure 14 illustrates the complexity of different detectionalgorithms in terms of number of operations in the caseof 4 times 4 spatial multiplexing systems using 64-QAM forthe 1st and 119894th iteration Similarly STS-SD presents morethan 90 reduction in the complexity compared to theMAP algorithm (12 times 10
9 MUL 11 times 109 ADD) We note
that the complexity of MMSE LC-MMSE and I-VBLASTslightly increases because the complexity of soft mapperand soft demapper increases with the constellation sizeMeanwhile the complexity of computing filter coefficientswill not be affected since the number of antennas is the sameWe notice also that the complexity of LC-K-Best decoderis approximately twice as much as that of LC-MMSE-ICequalizer However its complexity is about 40 lower than
Mobile Information Systems 17
Table 7 Complexity of LDPC decoder
ADDSUB LUT
CN update119872
sum
119895=1
(2119889119888119895
minus 1) +
119873
sum
119894=1
119889V119894 = 119872 (2119889119888
minus 1) + 119873119889V
119872
sum
119895=1
(2119889119888119895
) = 2119872119889119888
VN update119873
sum
119894=1
119889V119894 = 119873119889V mdash
428
164
332
109
392
152
2796
93
05
101520253035404550
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
STS-SD LC-K-Best
368
123 114
336
12396
05
10152025303540
LC-MMSE-IC
MULADDSUB
times102
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
64 64
148
40 48
8 8 16 4 4
020406080
100120140160
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
Figure 13 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
STS-SD (44 MUL and 45 ADD) It should be noted thateven thoughLC-MMSE-IChas a lower complexity it presentsa severe degradation of more than 2 dB in the case of 64-QAM in the Rayleigh channel andmore than 4 dB in realisticchannels (cf Section 61)
624 Complexity of Iterative Receivers In this section wecompare the complexity of the iterative receivers usingdifferent coding techniquesThe same simulation parametersas those used in the previous section are considered Weconsider a block length 119870
119887of 1024 for the turbo decoder and
codeword length 119873119887of 1944 in the case of LDPC decoder
which gives a block length slightly lower (5) than theturbo decoder case for 119877
119888= 12 Four outer iterations are
performed between the MIMO detectors and the channeldecoders The total number of iterations inside the LDPCdecoder and the turbo decoder is chosen to be 20 and 8iterations respectively because these number of iterations
were found sufficient for the convergence of both decoders(cf Sections 5 and 6)
The number of operations consumed by LDPC decoderand turbo decoder per information block length with coderates 119877
119888= 12 and 119877
119888= 34 is listed in Table 8 We notice
that the LDPC decoder requires 20 to 40 less operationsthan the turbo decoder Note that the decoding complexity ofturbo code is constant and does not depend on the code ratebecause all code rates are generated from the mother codingrate 119877 = 13 In contrast the complexity of LDPC dependson the code rate The decoding complexity decreases whenthe code rate increases
Figure 15 shows the computational complexity of theiterative receivers for one signal frame using both codingschemes and 16-QAM In the case of turbo decoder theLC-MMSE-IC equalizer presents the lowest computationalcomplexity in terms of MUL ADD However it requiresmore DIV and SQRT operations The complexity of STS-SD
18 Mobile Information Systems
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
Table 8 Complexity of turbo decoder (8 iterations) and LDPC decoder (20 iterations) in terms of number of operations
Turbo (8 iterations) LDPC (20 iterations)ADDSUB Max LUT ADDSUB LUT
119877119888
= 12 1856119870119887
704119870119887
704119870119887
554119870119887
287119870119887
119877119888
= 34 1856119870119887
704119870119887
704119870119887
371119870119887
189119870119887
6158
27563324
1105
5717
2624 2804
944
0
10
20
30
40
50
70
60
STS-SD LC-K-Best I-VBLAST MMSE
MULADDSUB
times102 1st iteration
5553
2536
1157
5157
2664
984
0
10
20
30
40
50
60
STS-SD LC-K-Best LC-MMSE-IC
MULADDSUB
times102
64 64
148
40 48
8 8 164 4
020406080
100120140160
STS-SD LC-K-Best I-VBLAST MMSE LC-MMSE-IC
DIVSQRT
1st iteration
ith iteration
ith iteration
Figure 14 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM for different detection algorithms in terms of the number ofoperations per symbol vector at the 1st and 119894th iteration
ismuch higher than the LC-K-Best decoder (about 60MULand 30ADD) Note that the more complexity brings only aperformance improvement of sim02 dB at a BER level of 1 times
10minus5 In addition the LC-K-Best decoder presents a reduced
complexity than I-VBLAST (20sim30 MUL 2sim5 ADDapproximately 50DIV and approximately 50 SQRT)Thereason is that I-VBLAST requires multiple matrix inversionsfor the first iteration Similar complexity results can beobserved in the case of LDPC decoder
By comparing the complexity of the receivers with bothcoding techniques we notice that the complexity of iterativereceiver with LDPC decoder is smaller than the complexitywith turbo decoder in terms of ADD Max and LUT opera-tions However both receivers present approximately similarcomplexity in terms of MUL DIV and SQRT
It is therefore worthy to compare the complexity of theiterative receiver with high modulation order and codingrate Figure 16 illustrates the computational complexity ofthe iterative receivers for one transmitted frame in 4 times 4
spatial multiplexing system with 64-QAM As shown inthe figure the complexity of the receiver based on STS-SD and LC-K-Best decoder increases significantly since thetree-search detection depends on the modulation order Thecomplexity of the receiver based on LC-MMSE-IC and I-VBLAST slightly increases compared to the case of 16-QAMdue to the small increases in the complexity of the softmapper and soft demapper Furthermore the complexity ofLC-MMSE-IC equalizer is much lower than the LC-K-Bestdecoder (sim55 MUL sim26 ADD) However LC-MMSE-ICpresents a significant degradation of about 2 dB in a Rayleighfading channel andmore than 4 dB inmore realistic channelsat the BER level of 1 times 10
minus4 compared to the LC-K-Bestdecoder (cf Section 6)
In addition Figure 16(b) shows that iterative receiverwithLDPC decoder presents low computational complexity interms of ADD LUTs However similar complexity of thereceiver with both coding techniques is observed in termsof MUL DIV and SQRT Since MUL and DIV are more
Mobile Information Systems 19
188
068 086057
362
257 263 239
072 072 072 072
005
115
225
335
4
MULADDSUB
Max LUT
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
081 081
373
235
010 010 036 020
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
(a) 4 times 4 16-QAM turbo decoder 119877119888= 12119870
119887= 1024
MULADDSUB
LUT
180
065081
055
218
116 123100
028 028 028 028
0
05
1
15
2
25
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
times106
005
115
225
335
4
078 078
351
224
0097 009760340 0195
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 16-QAM LDPC decoder 119877119888= 12 119873
119887= 1944 (119870
119887=
972)
Figure 15 Complexity of a 4 times 4 spatial multiplexing system with 16-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 12
complex thanADDMAX and LUTwe can conclude that thecomplexity of iterative receiver with both coding schemes iscomparable
From this evaluation we conclude that the performanceand the complexity of the iterative receiver with turbodecoder and LDPC decoder is highly comparable We shouldalso note that the turbo decoder is recommended for smallto moderate block lengths and coding rates Meanwhilethe LDPC decoder is more favored for large block sizesdue to their superior performance and lower complexityIn addition we see that the LC-K-Best decoder achieves agood performance-complexity trade-off compared to otherdetection algorithms Furthermore the LC-K-Best decoderperforms a breadth-first search that can be easily paralyzedand pipelined in hardware architecture as discussed in [1641] The LC-K-Best decoder can be also easily implementedand can provide a high and fixed detection rates for futurecommunication systems
7 Conclusions
The iterative receivers have recently emerged as very attrac-tive solutions for high data rate transmission in next gen-eration wireless systems In this paper an efficient itera-tive receiver combining MIMO detection based on K-Best
decoder with channel decoding namely turbo decoder andLDPC decoder has been investigated Several soft-input soft-output MIMO detection algorithms have been consideredin this work We analyzed the convergence of combiningthese detection algorithms with different channel decoders(turbo LDPC) using EXIT chart Based on this analysis weretrieved the number of innerouter iterations required forthe convergence of the iterative receiver Additionally weprovided a detailed comparison of different combinations ofdetection algorithms and channel decoders in terms of per-formance and complexity with real channel environmentsvarious modulation orders and coding rates Through theperformance and complexity evaluation we show that LC-K-Best decoder achieves a best trade-off between performanceand complexity among the considered detectors We showalso that the performance and the complexity of iterativereceivers with turbo decoder and LDPC decoder are highlycomparable Future work can include other aspects likeoptimization of the computational complexity in hardwarearchitecture estimation of the required memory conversionof the algorithm into a fixed point format and implementa-tion in real environments
Conflict of Interests
The authors declare that they have no competing interests
20 Mobile Information Systems
315
132087
058
482
325264 239
072 072 072 072
005
115
225
335
445
5
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
Max-LUT
times106
081 081
373
235
010 010035 020
005
115
225
335
4
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
(a) 4 times 4 64-QAM turbo decoder 119877119888= 34119870
119887= 1024
30
126082
055
333
183
124101
027 027 027 0270
051
152
253
35
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
MULADDSUB
LUT
times106
078 078
356
224
00976 00976 03416 019520
051
152
253
354
STS-SD LC-K-Best I-VBLAST LC-MMSE-IC
DIVSQRT
times104
(b) 4 times 4 64-QAM LDPC decoder 119877119888= 34 119873
119887= 1944 (119870
119887=
1458)
Figure 16 Complexity of a 4 times 4 spatial multiplexing system with 64-QAM of different detection algorithms with (a) LTE turbo decoder and(b) LDPC decoder 119877
119888= 34
Acknowledgments
Ming Liu is supported by the National Natural ScienceFoundation of China (no 61501022) and the Beijing JiaotongUniversity Foundation for Talents (no K15RC00040)
References
[1] E Telatar ldquoCapacity of multi-antenna Gaussian channelsrdquoEuropean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999
[2] H Vikalo and B Hassibi ldquoOn joint detection and decodingof linear block codes on Gaussian vector channelsrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3330ndash33422006
[3] C Berrou A Glavieux and P Thitimajshima ldquoNear Shannonlimit error-correcting coding and decoding turbo-codes 1rdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo93) vol 2 pp 1064ndash1070 IEEE GenevaSwitzerland May 1993
[4] C Douillard M Jezequel C Berrou A Picart P Didier andA Glavieux ldquoIterative correction of intersymbol interferenceturbo-equalizationrdquo European Transactions on Telecommunica-tions vol 6 no 5 pp 507ndash511 1995
[5] XWang andH V Poor ldquoIterative (turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999
[6] M Tuchler A C Singer and R Koetter ldquoMinimum meansquared error equalization using a priori informationrdquo IEEETransactions on Signal Processing vol 50 no 3 pp 673ndash6832002
[7] M Witzke S Baro F Schreckenbach and J HagenauerldquoIterative detection of MIMO signals with linear detectorsrdquo inProceedings of the 36th Asilomar Conference on Signals Systemsand Computers vol 1 pp 289ndash293 IEEE Pacific Grove CalifUSA November 2002
[8] L Boher R Rabineau and M Helard ldquoFPGA implementationof an iterative receiver for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 26 no 6 pp857ndash866 2008
[9] C Studer S Fateh and D Seethaler ldquoASIC implementation ofsoft-input soft-output MIMO detection using MMSE parallelinterference cancellationrdquo IEEE Journal of Solid-State Circuitsvol 46 no 7 pp 1754ndash1765 2011
[10] H Lee B Lee and I Lee ldquoIterative detection and decodingwith an improved V-BLAST for MIMO-OFDM systemsrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp504ndash513 2006
[11] J W Choi A C Singer J Lee and N I Cho ldquoImproved linearsoft-input soft-output detection via soft feedback successive
Mobile Information Systems 21
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
interference cancellationrdquo IEEE Transactions on Communica-tions vol 58 no 3 pp 986ndash996 2010
[12] B M Hochwald and S ten Brink ldquoAchieving near-capacity ona multiple-antenna channelrdquo IEEE Transactions on Communi-cations vol 51 no 3 pp 389ndash399 2003
[13] C Studer and H Bolcskei ldquoSoft-input soft-output single tree-search sphere decodingrdquo IEEE Transactions on InformationTheory vol 56 no 10 pp 4827ndash4842 2010
[14] E M Witte F Borlenghi G Ascheid R Leupers and H MeyrldquoA scalable VLSI architecture for soft-input soft-output singletree-search sphere decodingrdquo IEEE Transactions on Circuits andSystems II vol 57 no 9 pp 706ndash710 2010
[15] F Borlenghi E Witte G Ascheid H Meyr and A BurgldquoA 772Mbits 881bitnJ 90 nm CMOS soft-input soft-outputsphere decoderrdquo in Proceedings of the IEEE Asian Solid StateCircuits Conference (A-SSCC rsquo11) pp 297ndash300 Jeju SouthKorean November 2011
[16] Z Guo and P Nilsson ldquoAlgorithm and implementation of theK-best Sphere decoding for MIMO detectionrdquo IEEE Journal onSelected Areas in Communications vol 24 no 3 pp 491ndash5032006
[17] M Myllyla M Juntti and J R Cavallaro ldquoImplementationaspects of list sphere decoder algorithms for MIMO-OFDMsystemsrdquo Signal Processing vol 90 no 10 pp 2863ndash2876 2010
[18] D Patel V Smolyakov M Shabany and P G Gulak ldquoVLSIimplementation of a WiMAXLTE compliant low-complexityhigh-throughput soft-output K-best MIMO detectorrdquo in Pro-ceedings of the IEEE International Symposium on Circuits andSystems (ISCAS rsquo10) pp 593ndash596 Paris France June 2010
[19] M Mahdavi and M Shabany ldquoNovel MIMO detection algo-rithm for high-order constellations in the complex domainrdquoIEEE Transactions on Very Large Scale Integration (VLSI)Systems vol 21 no 5 pp 834ndash847 2013
[20] R El Chall F Nouvel M Helard and M Liu ldquoLow complexityk-best based iterative receiver for MIMO systemsrdquo in Proceed-ings of the 6th International Congress on Ultra Modern Telecom-munications and Control Systems and Workshops (ICUMT rsquo14)pp 451ndash455 IEEE Saint Petersburg Russia October 2014
[21] B Wu and G Masera ldquoEfficient VLSI implementation ofsoft-input soft-output fixed-complexity sphere decoderrdquo IETCommunications vol 6 no 9 pp 1111ndash1118 2012
[22] L Liu ldquoHigh-throughput hardware-efficient soft-input soft-output MIMO detector for iterative receiversrdquo in Proceedingsof the IEEE International Symposium on Circuits and Systems(ISCAS rsquo13) pp 2151ndash2154 IEEE Beijing China May 2013
[23] X Chen G He and J Ma ldquoVLSI implementation of a high-throughput iterative fixed-complexity sphere decoderrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 60no 5 pp 272ndash276 2013
[24] R E Chall F Nouvel M Helard and M Liu ldquoIterativereceivers combining MIMO detection with turbo decod-ing performance-complexity trade-offsrdquo EURASIP Journal onWireless Communications and Networking vol 2015 article 6919 pages 2015
[25] J J Boutros F Boixadera and C Lamy ldquoBit-interleaved codedmodulations for multiple-input multiple-output channelsrdquo inProceedings of the 6th International Symposium on SpreadSpectrum Techniques and Applications vol 1 pp 123ndash126 IEEESeptember 2000
[26] R G Gallager Low density parity-check codes [PhD thesis]MIT Press Cambridge Mass USA 1963
[27] J Hagenauer and P Hoeher ldquoA viterbi algorithm with soft-decision outputs and its applicationsrdquo in Proceedings of the IEEEGlobal Telecommunications Conference and Exhibition Commu-nications Technology for the 1990s and Beyond (GLOBECOMrsquo89) pp 1680ndash1686 Dallas Tex USA November 1989
[28] J Hagenauer E Offer and L Papke ldquoIterative decoding ofbinary block and convolutional codesrdquo IEEE Transactions onInformation Theory vol 42 no 2 pp 429ndash445 1996
[29] L R Bahl J Cocke F Jelinek and J Raviv ldquoOptimal decodingof linear codes for minimizing symbol error raterdquo IEEE Trans-actions on Information Theory vol 20 pp 284ndash287 1974
[30] P Robertson E Villebrun and P Hoeher ldquoA comparison ofoptimal and sub-optimal MAP decoding algorithms operatingin the log domainrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo95) vol 2 pp 1009ndash1013IEEE Seattle Wash USA June 1995
[31] T J Richardson and R L Urbanke ldquoThe capacity of low-density parity-check codes under message-passing decodingrdquoIEEE Transactions on InformationTheory vol 47 no 2 pp 599ndash618 2001
[32] R M Tanner ldquoA recursive approach to low complexity codesrdquoIEEE Transactions on InformationTheory vol 27 no 5 pp 533ndash547 1981
[33] E Agrell T Eriksson A Vardy and K Zeger ldquoClosest pointsearch in latticesrdquo IEEETransactions on InformationTheory vol48 no 8 pp 2201ndash2214 2002
[34] DWubben R Bohnke J Rinas V Kuhn and K D KammeyerldquoEfficient algorithm for decoding layered space-time codesrdquoElectronics Letters vol 37 no 22 pp 1348ndash1350 2001
[35] Y L C de Jong and T J Willink ldquoIterative tree searchdetection for MIMO wireless systemsrdquo IEEE Transactions onCommunications vol 53 no 6 pp 930ndash935 2005
[36] C-P Schnorr and M Euchner ldquoLattice basis reductionimproved practical algorithms and solving subset sum prob-lemsrdquo Mathematical Programming vol 66 no 2 pp 181ndash1911994
[37] DWubben R Bohnke V Kuhn andK-DKammeyer ldquoMMSEextension of V-BLAST based on sorted QR decompositionrdquo inProceedings of the 58th IEEE Vehicular Technology Conference(VTC rsquo03) vol 1 pp 508ndash512 IEEEOrlando Fla USAOctober2003
[38] E Zimmermann and G Fettweis ldquoUnbiasedMMSE tree searchdetection for multiple antenna systemsrdquo in Proceedings of theInternational Symposium onWireless Personel Mutimedia Com-munications (WPMC rsquo06) San Diego Calif USA September2006
[39] R Wang and G B Giannakis ldquoApproaching MIMO channelcapacity with reduced-complexity soft sphere decodingrdquo Pro-ceedings of the IEEE Wireless Communications and NetworkingConference (WCNC rsquo04) vol 3 pp 1620ndash1625 2004
[40] C Studer A Burg and H Bolcskei ldquoSoft-output spheredecoding algorithms and VLSI implementationrdquo IEEE Journalon SelectedAreas inCommunications vol 26 no 2 pp 290ndash3002008
[41] K-W Wong C-Y Tsui R S-K Cheng and W-H MowldquoA VLSI architecture of a K-best lattice decoding algorithmfor MIMO channelsrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo02) vol 3 pp III-273ndashIII-276 IEEE Phoenix Ariz USA May 2002
[42] S Chen T Zhang and Y Xin ldquoRelaxed K-best MIMO signaldetector design and VLSI implementationrdquo IEEE Transactions
22 Mobile Information Systems
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
on Very Large Scale Integration (VLSI) Systems vol 15 no 3 pp328ndash337 2007
[43] M Wenk M Zellweger A Burg N Felber and W FichtnerldquoK-Best MIMO detection VLSI architectures achieving up to424Mbpsrdquo in Proceedings of the IEEE International Symposiumon Circuits and Systems (ISCAS rsquo06) pp 1151ndash1154 IEEE May2006
[44] M Shabany and P G Gulak ldquoScalable VLSI architecture for K-best lattice decodersrdquo in Proceedings of the IEEE InternationalSymposium on Circuits and Systems (ISCAS rsquo08) pp 940ndash943Seattle Wash USA May 2008
[45] D L Milliner E Zimmermann J R Barry and G FettweisldquoA fixed-complexity smart candidate adding algorithm for soft-output MIMO detectionrdquo IEEE Journal on Selected Topics inSignal Processing vol 3 no 6 pp 1016ndash1025 2009
[46] JW Choi B Shim J K Nelson andA C Singer ldquoEfficient soft-input soft-outputMIMOdetection via improvedM-algorithmrdquoin Proceedings of the IEEE International Conference on Commu-nications (ICC rsquo10) pp 1ndash5 Cape Town South Africa May 2010
[47] P W Wolniansky G J Foschini G D Golden and RA Valenzuela ldquoV-BLAST an architecture for realizing veryhigh data rates over the rich-scattering wireless channelrdquo inProceedings of the URSI International Symposium on SignalsSystems and Electronics (ISSSE rsquo98) pp 295ndash300 IEEE PisaItaly September-October 1998
[48] I B Collings M R G Butler and M R McKay ldquoLowcomplexity receiver design for MIMO bit-interleaved codedmodulationrdquo in Proceedings of the IEEE International Sympo-sium on Spread SpectrumTechniques andApplications pp 12ndash16IEEE September 2004
[49] E Zimmermann and G Fettweis ldquoAdaptive vs Hybrid iterativeMIMO receivers based on MMSE linear and Soft-SIC detec-tionrdquo in Proceedings of the International Symposium on PersonalIndoor andMobile Radio Communications (PIMRC rsquo06) pp 1ndash5Helsinki Finland September 2006
[50] S T Brink ldquoConvergence behavior of iteratively decoded paral-lel concatenated codesrdquo IEEE Transactions on Communicationsvol 49 no 10 pp 1727ndash1737 2001
[51] J Hagenauer ldquoThe exit chartmdashintroduction to extrinsic infor-mation transfer in iterative processingrdquo in Proceedings of the12th European Signal Processing Conference pp 1541ndash1548Vienna Austria September 2004
[52] F Brannstrom L K Rasmussen and A J Grant ldquoConvergenceanalysis and optimal scheduling for multiple concatenatedcodesrdquo IEEE Transactions on Information Theory vol 51 no 9pp 3354ndash3364 2005
[53] E Sharon S Litsyn and J Goldberger ldquoAn efficient message-passing schedule for LDPC decodingrdquo in Proceedings of the 23rdIEEE Convention of Electrical and Electronics Engineers in Israelpp 223ndash226 IEEE Herzliya Israel September 2004
[54] M O Damen H E Gamal and G Caire ldquoOn maximum-likelihood detection and the search for the closest lattice pointrdquoIEEE Transactions on Information Theory vol 49 no 10 pp2389ndash2402 2003
[55] B Hassibi and H Vikalo ldquoOn the sphere-decoding algorithm IExpected complexityrdquo IEEE Transactions on Signal Processingvol 53 no 8 pp 2806ndash2818 2005
[56] J Jalden and B Ottersten ldquoParallel implementation of a softoutput sphere decoderrdquo in Proceedings of the 39th AsilomarConference on Signals Systems and Computers pp 581ndash585November 2005
