Riera-Palou and Femenias EURASIP Journal onWireless Communications andNetworking 2013, 2013:240http://jwcn.eurasipjournals.com/content/2013/1/240
RESEARCH Open Access
Channel-aware adaptive receivers for linearlyprecoded MIMO-OFDM systems withimperfect CSITFelip Riera-Palou* and Guillem Femenias
Abstract
Within the context of linearly precoded MIMO-OFDM (combination of multiple antenna techniques with multicarriertransmission schemes such as orthogonal frequency division multiplexing) systems with multiple-streamtransmission, maximum likelihood detection (MLD) has been shown to offer large performance gains when comparedto an all-linear setup (i.e., linear transmitter/receiver) when either perfect or imperfect channel state information at thetransmitter (CSIT) is available. Unfortunately, these gains come at the cost of a higher complexity. In particular, theincrease in computational cost is more significant when the receiver is designed to operate with soft information andeven more dramatic when, in order to optimise error rate performance, iterative decoding is allowed. In order toexploit the best features of each detection technique, this paper proposes a method to selectively choose thedetection strategy (ML or linear) for each individual subcarrier as a function of the instantaneous channel conditionsand CSIT accuracy. Numerical results show that a cautious and selective use of ML detection substantially reducescomplexity while still reaping most of the performance advantage.
1 IntroductionThe combination of multiple antenna techniques (so-called MIMO) with multicarrier transmission schemessuch as orthogonal frequency division multiplexing(OFDM), the so-called MIMO-OFDM architecture, isnow at the heart of most state-of-the-art wireless systemsand future standards [1-3]. In this context, techniquesthat exploit the availability of channel state information atthe transmitter (CSIT) have been intensively researched(see [4,5] for a review). It is well known that the capacity-achieving strategy when perfect CSIT is available is topre-cancel interference among simultaneously transmit-ted streams, a scheme usually referred to as dirty papercoding (DPC) [6]; however, its high computational costmotivates the need for simpler strategies. Palomar et al., intheir landmark paper [7], introduced a framework for theoptimisation of MIMO-OFDM systems with CSIT basedon linear processing at the transmitter and receiver. Theproposed scheme defines transmit and receiver filters thatare based on the singular value decomposition (SVD) of
*Correspondence: [email protected] Communications Group, Department of Mathematics and Informatics,University of the Balearic Islands, Mallorca, Illes Balears 07122, Spain
the whitened channel matrix and performs a distributionof the available power among the different transmit modesusing waterfilling in accordance with various performancemetrics. Further insight on this architecture was pro-vided in [8,9], where the diversity order performance wasanalysed for single- and multiple-stream configurations(i.e. spatial division multiplexing beamforming). Thesestudies showed that such schemes lose diversity whenincreasing the number of transmitted streams as perfor-mance becomes dominated by the worst employed spatialtransmission mode. Very recently, it has been shown in[10] that full diversity can be restored by incorporating alinear transformation at transmission spreading the sym-bols to be transmitted over the available spatial modes.Unfortunately, this diversity advantage comes at the costof having to rely on joint maximum likelihood detection(MLD) at the receiver.Most of these results assumed perfect CSIT, which is
a rather optimistic hypothesis in practical deployments.Channel feedback delay and quantization noise are typ-ical impairments affecting the quality of CSIT, whoseeffects should be accounted for. To this end, [11] incor-porated channel knowledge imperfections in the design
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of a linear transmitter/receiver architecture that, as theCSIT approaches perfection, converges towards the solu-tion of [7]. A related work by Sengul et al. [12] proposesa codebook construction methodology based on a Lloydquantizer design that aims at the improvement of therobustness against imperfect CSIT in linearly precodedbit-interleaved coded modulation (BICM) systems whilestill relying on linear filters at the receiver side. In [13],precoding strategies combined with forward error cor-rection were considered but again limiting the context tothat of linear detectors. Remarkably, it should be notedthat, under imperfect CSIT, the optimisation of error ratemetrics requires MLD-based reception.The use of MLD in combination with linear precod-
ing has been extensively studied in [14] and [15] underperfect and imperfect CSIT, showing that large reduc-tions in the bit/packet error rate (BER/PER) are possibleat the cost of an increased receiver complexity as evensmart implementations (i.e. sphere decoding [16]) arecomputationally demanding at low signal-to-noise ratios(SNRs), where practical systems usually operate [17]. Toaddress this downside, this paper proposes the selectiveand careful application of MLD only under very specificconditions, which depend on the specific channel real-isation and CSIT accuracy, while otherwise relying onlinear detection. The introduced technique is shown tobe effective with various architectures, namely hard-, soft-and iteratively decoded receivers. This scheme is spe-cially appropriate for scenarios where the channel and/orCSIT accuracy may vary widely from packet to packet.As an illustrative example of this type of scenario, thispaper considers wireless local area networks (WLANs)based on the IEEE 802.11n standard [18], whose multiple-access policy based on carrier sense multiple access withcollision avoidance (CSMA/CA) causes substantial varia-tions in the accuracy of the available channel informationat the transmitter. It is worth mentioning that a relatedidea, but in the context of MIMO systems without CSITand restricted to the 2 × 2 MIMO setup, was intro-duced in [19], where the detection strategy selection wasbased on the condition number of the channel correlationmatrix resulting in the utilisation of linear and ML detec-tion for well-conditioned and ill-conditioned channels,respectively.The rest of the paper is organised as follows: Section 2
introduces the system model under consideration includ-ing a description of the assumptions regarding the channelmodel and CSIT accuracy. Section 3 begins by review-ing the two classic detectors at hand, minimum meansquare error (MMSE) and MLD, within the context ofthe considered scenario and subsequently introduces thechannel-aware adaptive detector. In Section 4, the adap-tive detector concept is revisited within the framework ofsoft and iterative detection strategies. Numerical results
are presented in Section 5 illustrating the benefits ofthe channel-aware adaptive detector. Finally, the mainoutcomes of this work are recapped in Section 6.This introduction concludes with a brief notational
remark. Vectors and matrices are denoted by bold lowercase and bold upper case letters, respectively. The super-script (·)H denotes the complex transpose (Hermitian)of the corresponding variable. The symbol Ik denotesthe k-dimensional identity matrix, whereas D(x) is usedto represent a (block) diagonal matrix having x at itsmain (block) diagonal and [A]i,j serves to indicate the(i, j)-element of matrix A.
2 SystemmodelA MIMO-OFDM architecture is considered where thetransmitter and receiver are equipped with NT and NRantennas, respectively, which are capable of simultane-ously transmitting Ns ≤ min(NT ,NR) data streams. Theavailable system bandwidth is exploited by means of Ncsubcarriers out of which Nd are used to carry data and Npare destined to pilot signals and guard bands.2.1 Transmitter processingFollowing the usual processing steps of BICM systems,incoming information packets are first channel encodedand possibly punctured to satisfy prescribed rate con-straints, and the resulting bits are then distributedamong Ns streams corresponding to different spatialbranches. On each spatial branch, bits are interleaved andmapped onto modulation symbols drawn from an M-ary modulation alphabet, resulting in the set of symbolstreams
{s1, · · · , sNs
}. Each spatial stream is then organ-
ised into segments of Nd symbols that will eventuallybecome OFDM symbols (with the addition of pilot/nullsubcarriers) (Figure 1).The mapping from information to transmit symbols
(i.e. precoding) on subcarrier q at discrete time instant nis carried out as
x[q, n]= W[q, n] s[q, n] , (1)
where W[q, n], with dimensions NT × Ns, represents theprecoding matrix and s[q, n] = (
s1[q, n] · · · sNs [q, n])T ,
with si[q, n] denoting the symbol corresponding to the ithstream to be transmitted on the qth subcarrier at timeinstant n. Finally, the precoded symbols are supplied to anOFDMmodulator consisting of an IFFT plus the additionof a cyclic prefix (CP).
2.2 Channel modellingThe channel between an arbitrary pair of Tx and Rxantennas is assumed to be frequency-selective with ascenario-dependent power delay profile common to allTx-Rx pairs. Let us denote byH[q, n] thematrix represent-ing the channel frequency response on the qth subcarrier
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Figure 1 Transmitter block diagram for linearly precoded system.
H[q, n]=⎛⎜⎝
h1,1[q, n] · · · h1,NT [q, n]...
...hNR,1[q, n] · · · hNR,NT [q, n]
⎞⎟⎠ , (2)
where an arbitrary entry hi,j[q, n] (1 ≤ i ≤ NR, 1 ≤ j ≤NT ) corresponds to the frequency response on subcarrierq of the channel linking Tx antenna j and Rx antenna i.It is assumed that the entries of H[q, n] are uncorrelateda(i.e. Tx/Rx antennas are sufficiently spaced).Channel estimation at the receiver is considered accu-
rate enough so as to render any CSI error at the Rx side(CSIR) negligible. In contrast, CSIT is considered to beimperfect in such a way that [11]
H[q, n]= ρ[n] H̄[q, n]+√1 − ρ[n]2�[q, n] , (3)
where H̄[q, n] represents the channel mean known atthe transmitter (estimated channel), �[q, n] denotes thechannel estimation noise whose entries are CN (0, 1), andρ[n]∈ [0, 1] can be a packet-dependent random variableeffectively modelling the CSIT accuracy for the currentpacket, which is also known at the receiver side.
2.3 Reception equationFigure 2 depicts a generic receiver based on hard decod-ing. Reception begins with the standard OFDM demodu-lation consisting of the CP removal and FFT processing.Perfect Tx/Rx synchronisation and a CP length exceed-ing the duration of the channel impulse response areassumed, thus guaranteeing that consecutive OFDM sym-bols do not suffer from inter-block interference. Underthese conditions, the detection procedure works on anOFDM symbol basis, thus allowing us to drop the time-related index n from subsequent equations (e.g. s[q, n]→
s[q]). The received baseband samples on subcarrier q,denoted by r[q]= [
r1[q] · · · rNr [q]]T with ri[q] represent-
ing the received sample on the ith receive antenna, for anarbitrary OFDM symbol are given by
r[q]= A[q] s[q] + υ[q] , (4)
where A[q] = H[q]W[q], and the NR × 1 vector υ[q]corresponds to the noise samples affecting the qth sub-carrier, which are assumed to be i.i.d. and drawn from azero-mean complex Gaussian distribution with varianceσ 2v . It is assumed that, on average, each subcarrier has
unit energy available to transmit Ns symbols and that thechannel frequency response is normalised so that the aver-age signal-to-noise ratio per subcarrier can be defined asEs/N0 = 1/(Nsσ 2).
3 Channel-aware robust detectionIn [11] a linear architecture is proposed able to robustlycope with CSIT imperfections. In particular, a practi-cal (uncoded/hard decoded) BER minimisation approachconsists of a precoding filter defined by
W[q]= U[q] � [q]C, (5)
where U[q] has as columns the eigenvectors of R̄q =ρ2H̄[q]H H̄[q] +Ns
1σ 2v
corresponding to its Ns largest
eigenvalues, and �[q]= D(ω1q · · ·ωNsq
)is the power allo-
cation matrix whose coefficients can be found optimis-ing a prescribed objective metric [7]. The matrix C is a(subcarrier-independent) unitary transform that spreadsthe incoming symbols among the different spatial modes.It has been recently shown in [10] that choosing Cto be the product of a unitary transform (e.g. Fourier,Hadamard) and a constellation rotation, in the form of a
Figure 2 Receiver block diagramwith hard detection.
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diagonal matrix with different phase factors, maximisesdiversity and leads to optimum performance in terms ofBER.
3.1 Linear and non-linear detectorsGiven the received samples in (4), the optimum linearMMSE receive filter is given by [11]
G[q]= (A[q]AH [q]+ σ 2INR
)−1A[q] , (6)
allowing (hard) symbol estimates to be obtained as
s̃[q]MMSE = GH [q] r[q] . (7)
This detector is also BER optimal when two conditionsare met:
1. The CSIT is perfect.2. The rotation matrix C in (5) is diagonal.
Under these two conditions, the overall precoder-channnel-detector chain is perfectly diagonalized, thusallowing symbol-by-symbol detection without any per-formance penalty. Nevertheless, note that condition 2,despite simplifying the detection procedure, inherentlyinduces a diversity loss as it implies that, in loaded setups(Ns = min{NT ,NR}), some of the symbols are transmit-ted on weak spatial modes that will dominate the errorperformance [8]. Very recently, [14] and [15] have studiedwhat gains are achieved when one or both of these condi-tions are not fulfilled and the receiver linearity constraintis neglected, thus allowing the application of MLD. In thiscase, symbol estimates are given by
s̃[q]ML = argmins[q]
∣∣r[q]−A[q] s[q]∣∣2 . (8)
Results in [14,15] show that, in fully loaded configura-tions, (8) is very advantageous over (7) in terms of BER,although this comes at the cost of an increased receivercomplexity, even when employing efficient implementa-tions such as sphere decoding. Regardless of the detectionmethod, either MMSE or ML, estimated symbols are thendemodulated, and the corresponding bits, subsequentlyde-interleaved, (spatially) de-parsed and finally suppliedto a Viterbi decoder to obtain an estimate of the transmit-ted packet.
3.2 Adaptive detectorAssuming that the receiver has knowledge of the precod-ing matrix used by the transmitterb, a detection strategydecision can be made based on the instantaneous channel
realisation and specific CSIT accuracy. To this end, let thelinear receiver form the overall processing matrix,
where the diagonal terms λ1[q] , · · · , λNs [q] correspond tothe eigenvalues of R̄q and form the set of accessible spa-tial modes available for transmission. There is an intimateconnection between the two conditions guaranteeing theoptimality of MMSE detection and the structure of (9):
• When conditions 1 and 2 hold, it is obvious thatβi,j[q]= 0 ∀i, j and MMSE detection is optimum.
• When condition 1 holds and condition 2 does nothold, the magnitude of the interfering terms βi,j[q]depends on the conditioning ofH[q]. If the matrix iswell conditioned, MMSE will perform well, but if it isnot, MLD will result in a significant advantage.
• When condition 1 does not hold and condition 2holds, it will depend on the actual realisation ofparameter ρ. If ρ � 1, the overall processing matrixwill be virtually diagonalized, making MMSEdetection optimal. In contrast, the further away ρ isfrom 1, the more significant interfering terms βi,j[q]will become, thus requiring MLD for acceptableperformance.
• When conditions 1 and 2 do not hold, the detectionstrategy selection will depend on both the channelmatrix conditioning and the specific CSIT accuracy.
To find a decision criterion able to determine the mostappropriate detection strategy in light of the instanta-neous conditions of the system, let us define SNRi[q] as
SNRi[q]= |λi[q] |2σ 2n
. (10)
Similarly, the signal-to-interference plus noise ratio(SINR) can be defined as
SINRi[q]= |λi[q] |2σ 2n + ∑Ns
j=1,j �=i |βi,j[q] |2. (11)
Clearly, on those subcarriers where SNRi[q]�SINRi[q], there is a strong indication of significant inter-fering terms (either because of poor channel conditioningand/or mismatched transmit and receive filters due toimperfect CSIT) that would favour the application ofMLD. With this observation in mind, Algorithm 1 can beused to decide which detection scheme should be selectedon the qth subcarrier. To this end, the algorithm evaluatesexpressions (10) and (11) on each subcarrier for each
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spatial mode and, utilising the mapping F : Ns → ,derives two metrics, one for the overall SNR and anotherone for the overall SINR, on which the detection decisionfor that subcarrier will be based. Potential mapping func-tions to be used are the mean, minimum or maximumof the incoming Ns-long vector. Parameter α ∈ [0, 1] inAlgorithm 1 is used to allow a certain amount of interfer-ence to be present yet still relying on MMSE detection.It effectively acts as a complexity-performance tradingknob: as α → 0, the more interference is allowed and themore often the system relies on MMSE detection (lowercomplexity and poorer performance than with ML). Twoimportant remarks are in place:
• The subcarrier-based nature of the algorithm is to beemphasised. Most likely, for a given packet, some ofthe subcarriers will be linearly detected while otherswill require the use of MLD.
• Sphere detection-based MLD usually starts thesearch procedure using the zero forcing (ZF) solutionas the centre of the sphere. However, if an estimate ofthe noise power σ 2 is available, centering the searcharound the MMSE solution is computationallyadvantageous [20]; thus, the computation of (6), evenin the case of eventually relying on MLD, still plays arole.
Algorithm 1 Detection decision procedure.Available information at Rx:W[q] ,H[q] , σ 2
n ,α.1) MMSE matrix computation G[q] using (6).2) Form SNRi[q] and SINRi[q] using (10)-(11).3) Decision criterion
θSINR[q]= F (SINR1[q] , · · · , SINRNs [ q]
)θSNR[q]= F (
SNR1[q] , · · · , SNRNs [ q])
if θSINR[q]< α θSNR[q] thenDetection on subcarrier q using ML.
elseDetection on subcarrier q using MMSE.
end if
4 Iterative soft detectionDespite the importance of hard decoding in its own right,most practical deployments are based on the use of soft-based decoding principles. Consequently, it is importantto consider the performance of the proposed adaptivedetection scheme when the component detectors extractsoft information, typically in the form of log-likelihoodratios (LLRs), from the received samples. Furthermore,soft detectors are often able to operate iteratively fol-lowing turbo receiver design principles. In this case, theMIMO detector and channel decoder exchange (soft)information back and forth with the corresponding LLRsbecoming more reliable at each iteration [21]. The next
subsections describe two popular soft-based detectionschemes, one based on MLD and another one based onMMSE, and the iterative extension within the context ofthe considered setup.
4.1 MLD-based soft detectionFor the case of ML detection based on the sphere decoder,the authors in [21] introduced the list sphere decoder(LSD) that not only renders the most likely (hard) estimates̃[q]ML but also provides a list of the closest candidatepoints to the ML solution. This list enables the deriva-tion of soft information in the form of LLR for each bit.To this end, the transformation s[q]= M (b) is defined asthe modulation mapping to arrive at symbol vector si[q]from the corresponding bitsc b = (
b1 b2 . . . bNb
)T whereNb = log2M. Making use of the max-log approximation,the LLR for a given bit bp (belonging to an arbitrarysubcarrier q and stream i) can be approximated by [21]
LMLi [q]
(si[q] , bp
)
≈ 12
maxb∈Bp,+1
{− 1
σ 2n
‖r[q]−A[q]M(b)‖2}
− 12
maxb∈Bp,−1
{− 1
σ 2n
‖r[q]−A[q]M(b)‖2},
(12)
where the characters Bp,+1 and Bp,−1 represent the setsof 2Nb−1 bit vectors whose pth position is a ‘+1’ or ‘−1’,respectively. Moderate values of M and/or Ns make thesets Bp,+1 and Bp,−1 extremely large, making the searchin (12) computationally challenging. To address this issue,the LSD limits the search to the sets B̂p,+1 = Bp,+1 ∩ Cand B̂p,−1 = Bp,−1 ∩ C where C is the set contain-ing the bit vectors corresponding to the Ncand candidatescloser, in a Euclidean sense, to the received samples, i.e.C = {b1, . . . , bNcand
}where bn = M−1(s̃[c][q] ) with{s̃[1][q] , . . . , s̃[Ncand][q]}being the Ncand group candi-
dates for which the Euclidean distance ‖r[q]−A[q] s[q] ‖2is smallest.
4.2 MMSE-based soft detectionIn order to derive soft estimates from theMMSE-detectedsamples, the procedure described in [22] is adapted to thesituation at hand. To this end, let us define the post-MMSEreceive filter SNR for an arbitrary subcarrier q and streami as
SNRi[q]MMSE = 1[(1σ 2A[q]AH [q]+INR
)−1]i,i
− 1.
(13)
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Based on [22], the LLR for the in-phase bit on the pthposition of the symbol stream i is given by
LMMSEi [q]
(si[q] , bI,p
) = SNRi[q]MMSE
4DI,p, (14)
where DI,p is given by the mappings defined in [22] (14 to18). LLRs for the bits in quadrature are computed using ananalogous procedure.Note that the extraction of the soft information is much
more involved in the case of MLD-based processing, evenwhen employing the lower-complexity LSD, than for theMMSE-based detector. As shown in Figure 3 and ignor-ing for now the iterative processing (shaded region), theresulting LLR streams
{LD11 , · · · ,LD1
Ns
}, whose entries are
given by LMLi or LMMSE
i depending on the detectionprocedure employed for the corresponding subcarrier,are subsequently de-segmented, de-interleaved, de-parsedand de-punctured to form the coded LLR stream that isfinally supplied to a Viterbi decoder to yield the estimatedinformation bits.
4.3 Channel-aware iterative soft detectionFurther performance improvements in the form of lowererror rates can be achieved if the detector and channeldecoder are allowed to exchange information, speciallywhen the detector is based on ML detection principles[21]. In fact, it has been observed that when detectionrelies on linear processing techniques such as MMSE,the benefits of iterative reception become rather marginal[23]. Consequently, in this work, the application of itera-tive processing is limited to those cases where MLD hasbeen selected as the preferred detection strategy.As shown in Figure 3, each (subcarrier-based) detector
operates in accordance with Algorithm 1 to decide whichdetection strategy, MLD or MMSE, should be used and
computes the corresponding LLR for each bit using (12) or(14), respectively. The de-segmentation process is then incharge of collecting the LLR values computed for the suc-cessive OFDM symbols forming a packet/stream result-ing in the LLR streams
{LD11 , · · · ,LD1
Ns
}. These LLRs,
after subtracting any a priori knowledge available fromprevious iterations, give rise to the extrinsic informa-tion
{LE1
1 , · · · ,LE1Ns
}, which after suitable de-interleaving
and spatial de-parsing, results in the input stream to themaximum a posteriori (MAP) decoder (LA2
1 ). The MAPdecoder has a double output: on one hand, an estimate ofthe information symbols, and on the other hand, a refinedversion of the input LLRs. This latter output, LD2
1 , aftersubtracting already known information (LA2
1 ), results inthe extrinsic information LE2
1 to be fed back to the detec-tion stage. To this end, signal LE2
1 is suitably parsed andinterleaved resulting in the sequences
{LA1
1 , · · · ,LA1Ns
}forming the a priori information for the next turbo iter-ation. Note that only the LLRs corresponding to thosesubcarriers that have been detected using MLD are fedback to the detector (denoted in Figure 3 by
{LA1
i
}ML
)while no information is fed back to the MMSE-detectedsubcarriers.
5 Numerical results5.1 Simulation setupThe simulation environment has been defined in accor-dance with specifications from the IEEE 802.11n archi-tectures [18], considering a setup with NT = NR = 4antennas transmitting Ns = 4 streams. The system oper-ates on a bandwidth of B = 20 MHz using Nc = 64subcarriers out of which Nd = 52 are used for datatransmission and the rest are devoted to pilot signallingand guard bands. For all simulations, transmission modes
Figure 3 Channel-aware turbo receiver.
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with either quadrature phase shift keying (QPSK) or 16-QAM modulation and a 1/2-rate convolutional channelcoder with generator polynomials g = [133, 171]8 havebeen employed. Full cyclic prefix is used in order to guar-antee the avoidance of interference among successivelytransmitted OFDM symbols. Power allocation matricesare computed according to the ARITH-MSE criterion in[7] for hard decoding and uniform power allocation forsoft/iterative decodingd. Two different spatial spreadingmatrices have been considered, C = INs (no spreading)and C = �Ns (full spreading), with INs and �Ns denot-ing the identity and rotated Walsh-Hadamard matrices ofdimension Ns, respectively.Interestingly, IEEE 802.11-based systems are a represen-
tative scenario where the CSIT accuracymay (widely) varyover a short time frame. This is due to the channel con-tention mechanism that, based on CSMA/CA, causes thetime span between the reception of channel-related feed-back at Tx and its utilisation to fluctuate on a packet basisand, moreover, to make it heavily dependent on the num-ber of active users in the system. Note that when usersenter or exit the system, the average delay in using theacquired CSIT for the rest of the active users is likely tovary, thus effectively implying a degradation or improve-ment in the CSIT accuracy. For the results shown here, thechannel, generated following the specifications in [24], isassumed to remain static over the duration of a packet andvary independently from packet to packet (block fading).Without loss of generality, it is assumed in (3) that
ρ = 1 − ϕ where ϕ is a random variable with a gammaprobability density function�(μ, 1) truncated to the inter-val [0, 1]. In particular, the results here are presented forμ = 0.015, μ = 0.37 and μ = 1.31, which lead toaverage values of ρ̄ = 0.99, ρ̄ = 0.8 and ρ̄ = 0.5, respec-tively. Figure 4 depicts histograms for 2,500 realisations
(i.e. packets) of the resulting ρ parameter for the threedifferent values of μ. It can clearly be appreciated that,indeed, a value of μ = 0.015 results in a very accurateCSIT for most of the frames, whereas when μ = 0.37or μ = 1.31, the corresponding ρ values clearly suggestthat a significant proportion of packets are transmittedwith a rather imperfect CSIT. Note that the generationof ρ based on the gamma distribution allows the mod-elling of various network operating conditions just byadjusting a single parameter (μ). Algorithm 1 parame-ters for the hard-decoded setup have been chosen to beα ∈ {0.25, 0.5, 0.75}whileF(x) = min(x), and for the soft-decoded setup, α ∈ {0.75, 0.85, 0.95} and F(x) = min(x).Lastly, for the iterative decoding configuration, Algorithm1 is configured with F(x) = min(x) and α = 0.975.The different choice of α for the hard, soft and iterativedecoding is due to the very different SNR levels each ofthese configurations operates in that affects what can beregarded as a tolerable level of interference or not. Theuse of other mapping functions such as F(x) = max(x)or F(x) = mean(x) has also been tested, but the corre-sponding results do not differ substantially from the onespresented next using F(x) = min(x).
5.2 Hard-decoded resultsFigure 5 presents results for the non-spread setup. Theleft plot shows the PER performance when very accurateCSIT is available, and as expected under these conditions,the linear receiver in (6) virtually diagonalises the over-all processing chain, thus making linear detection nearlyoptimal. Only for those very few cases where CSIT error issignificant, MLD provides an advantage overMMSE, a sit-uation leading to the rather small gain observed betweenthe performance of both fixed detectors. The performancefor the adaptive detector following Algorithm 1 remains
0 0.5 10
500
1000
1500
2000
2500
ρ
Fre
quen
cy o
f occ
uren
ce
μ=0.015
0 0.5 10
200
400
600
800
1000
ρ
Fre
quen
cy o
f occ
uren
ce
μ=0.37
0 0.5 10
20
40
60
80
100
ρ
Fre
quen
cy o
f occ
urre
nce
μ=1.31
Figure 4 Histogram of ρ derived from 2,500 independent draws when μ = 0.015 (left),μ = 0.37 (centre) andμ = 1.31 (right).
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Figure 5 PER and complexity for setup without spatial spreading (C = INs ) using hard decoding. QPSK modulation.
unaltered regardless of the value of α. The top-right plotin the same figure shows the percentage of utilisationof each detector for the particular case of α = 0.75,and remarkably, Algorithm 1 overwhelmingly chooses thelinear approach for any SNR level. When CSIT qual-ity diminishes, differences among the PER performanceof the various detectors appear, as shown in the centralplot in Figure 5. There are two noticeable facts regardingthe performance of MLD: firstly, it now clearly outper-forms linear detection, and secondly, it improves upon theresults obtained using very accurate CSIT. This somewhatcounterintuitive effect, already observed and discussed in[15], is caused by the absence of spatial spreading: themis-match between Tx and Rx due to a lousy CSIT causessignificant interfering terms to appear in (9) that effec-tively act as a form of spreading that the MLD can exploit,resulting in a 2-dB gain over linear processing. The adap-tive detector, depending on the value of α, exhibits vari-ous degrees of performance. For the particular choice ofα = 0.75, it basically achieves optimum performance
while, as shown on the bottom-right plot of Figure 5, itonly triggers the use of MLD for about 40% to 50% of thedetector invocations with the rest of the times relying onlinear processing.Results in Figure 6 correspond to a setup employing
rotated Walsh-Hadamard spatial spreading. In this case,either with accurate or inaccurate CSIT, MLD clearly out-performs linear detection by up to 4 dB (left and centralplots in Figure 6). The adaptive detector is seen to yielddifferent error rates depending on the chosen α. For theparticular choice of α = 0.75, the attained PER is within0.5 dB of the optimal solution (fixedMLD) while requiringonly the use of MLD for about 40% of the detector invo-cations when CSIT is accurate and for around 60% whenthis is rather inaccurate.
5.3 Soft-decoded resultsFor the soft-decoded case, only non-spread configurationsare considered (C = INs ) as it has been shown in [14,25]that, in soft-based BICM systems, spatial spreading leads
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Figure 6 PER and complexity for setup with rotatedWalsh-Hadamard spatial spreading (C = �Ns ) using hard decoding. QPSK modulation.
Riera-Palou and Femenias EURASIP Journal onWireless Communications and Networking 2013, 2013:240 Page 9 of 11http://jwcn.eurasipjournals.com/content/2013/1/240
to a PER performance degradation. PER performance andcomplexity results are shown in Figure 7 for the threeconsidered levels of CSIT accuracy when employing soft-based decoding. Focusing on the PER performance, it canbe observed that the more inaccurate the CSIT is, thebigger the difference between MLD and MMSE detec-tion becomes. In particular, all detectors achieve similarperformance for μ = 0.015 (top-left plot in Figure 7)whereas for μ = 0.37, and even more markedly whenμ = 1.31, the performance gap between ML and MMSEis somewhere between 2 and 3 dB for the typically rel-evant range of PER values (10−3 to 10−1). The adaptivedetector behaves as expected: for near-perfect CSIT, sim-ilar PER performance was obtained independent of α
(α ∈ {0.75, 0.85, 0.95}), while under imperfect CSIT, theperformance (and complexity) of the adaptive detector isgreatly influenced by the particular choice of α. The bot-tom plots in Figure 7 represent the percentages of detectorutilisation when employing Algorithm 1 with α = 0.95 forthe three values of μ. Notice that large values of α suchas this one strive for performance optimisation. As antic-ipated, when CSIT is nearly perfect, the adaptive detectorrelies almost exclusively on MMSE detection. In contrast,as CSIT accuracy degrades, the adaptive detector tendsto favour the more frequent use of MLD to the pointthat for very poor-quality CSIT information, the adaptivedetector almost invariably triggers the use of MLD. Nev-ertheless, note that, although not shown here and as inthe hard-decoded scenario, a lower value of α would leadto significant computational savings (more frequent use ofMMSE for μ = 0.37 or μ = 1.31) at the cost of a slightdegradation in PER performance.
5.4 Iteratively decoded resultsFigure 8 shows results when iterative decoding is applied.For this scenario, the system was configured to use16-QAM modulation, no spatial spreading was in use(C = INs ) and the adaptive detector’s threshold was setto α = 0.975. Results are shown for two different degreesof statistical CSIT quality, μ = 0.015 and μ = 1.31, cor-responding to very accurate and rather inaccurate chan-nel knowledge, respectively. The first remarkable pointto note is that, as noted in previous results, when thetransmitter can rely on a good channel estimate (μ =0.015), there are no very large differences between the(non-adaptive) ML and MMSE soft detectors, with onlya marginal benefit for the ML-based receiver as shownin the left plot in Figure 8. When iterative detection isallowede, the ML detector is able to somewhat improveperformance over its non-iterative counterpart. Nonethe-less, it is remarkable that the SNR gap between thelow-complexity (non-iterative) MMSE-based soft detec-tor and the much more complex iterative ML receiver isless than 1 dB. When the detection strategy on each sub-carrier is selected according to Algorithm 1, it can beclearly observed that for the vast majority of cases, MMSEdetection is selected (see top-right plot in Figure 8) withML resulting in the chosen strategy only for those fewcases where there was a considerable mismatch betweenthe CSIT and the true channel. Consequently, the perfor-mance of the adaptive detector lies in the narrow groundbetween that of the fixed receivers. As shown in the figure,the adaptive detector was tested with and without the pos-sibility of iterating, and as with the fixed ML decoder, itwas observed that iterating resulted in a rather marginal
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Figure 7 PER and complexity for setup with no spatial spreading (C = INs ) using soft decoding. QPSK modulation.
Riera-Palou and Femenias EURASIP Journal onWireless Communications and Networking 2013, 2013:240 Page 10 of 11http://jwcn.eurasipjournals.com/content/2013/1/240
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Figure 8 PER and complexity for setup with no spatial spreading (C = INs ) using iterative decoding. 16-QAMmodulation.
improvement. Note that as mentioned in Section 4.3,when using the channel-aware iterative detector, infor-mation feedback is only conducted for those subcarriersemploying ML detection.When the statistical CSIT quality is considerably
degraded (μ = 1.31), results are significantly different.As it can be observed in the middle plot of Figure 8,and in line with results in the previous subsection, non-iterative ML detection provides a gain of nearly 3 dB withrespect to MMSE. Furthermore, when the ML receiver isallowed to iterate, a further 1- to 1.5-dB gain is achieved.When using channel-aware detection, it is observed thatAlgorithm 1, either with or without iterations, rightlychooses to rely on ML detection owing to the rather largeinterfering terms the receiver observes when evaluating(10) and (11).
5.5 An important remarkIt is worth emphasising that the merits of the adaptivedetector should be valued by globally appreciating theresults on each of the considered scenarios (hard, soft anditerative decoding): a single configuration for each typeof receiver (α = 0.75,F(x) = min(x) for hard decod-ing, α = 0.95,F(x) = min(x) for soft decoding andα = 0.975,F(x) = min(x) for iterative decoding) leadsto a strategy able to attain virtually optimum PER perfor-mance while potentially offering a very significant com-plexity reduction with respect to the full use of MLD. Inother words, Algorithm 1 provides the receiver with thecapability of distinguishing when MLD will be effectiveand when MMSE will suffice. This scheme can thereforebe very attractive in those scenarios where the qual-ity of CSIT may vary over time such as it occurs intoday’s WLAN environments depending on the num-ber of users in the system or changes in the environ-ment. Furthermore, note that the parameter α acts as aperformance/complexity trading knob, thus enabling the
reconfiguration of the system as a function of, for instance,the available battery power or required processing latency.
6 ConclusionsThis paper has proposed an adaptive detection techniquethat allows the receiver of a MIMO-OFDM linearly pre-coded system to toggle between the use of MMSE andMLD depending on the CSIT accuracy and/or channelconditioning. The introduced technique works on a per-subcarrier basis and is compatible with different receiverarchitectures, namely hard, soft and iterative decodingschemes. Numerical results have shown that regardless ofthe receiver setup, the adaptive detector is able to dis-tinguish the system conditions that allow MLD to boostperformance from those where the much simpler MMSEdetector would perform (near) optimally.
EndnotesaThe incorporation of antenna correlation effects to the
current system model is trivial; however, it unnecessarilycomplicates notation without providing any furtherinsight or significantly altering numerical results.
bThis is a realistic assumption since the receiver shouldbe aware of the last CSI information sent to thetransmitter. Alternatively, if the transmitter sends pilotsymbols through the precoder (and channel), the receivercan also deduct the precoding filter used in transmission.
cTo simplify the notation, the subcarrier and streamindices are skipped when referring to the bits.
dThe (possibly iterative) utilisation of soft informationat the receiver suggests using capacity-based measures tooptimise the power allocation. Unfortunately, underimperfect CSIT, no closed-form solutions are availableand power allocation solutions require convex numericaloptimisation procedures (see [5] for a detailed discussionon this issue).
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eFor the results shown here, only two iterationsbetween detector and decoder were allowed as it wasobserved that further iterations did not bring along anysignificant performance benefit.
Competing interestsThe authors declare that they have no competing interests.
AcknowledgementsWork funded by MINECO and FEDER under project AM3DIO (TEC2011-25446),Spain.
Received: 14 May 2013 Accepted: 24 September 2013Published: 7 October 2013
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doi:10.1186/1687-1499-2013-240Cite this article as: Riera-Palou and Femenias: Channel-aware adaptivereceivers for linearly precoded MIMO-OFDM systems with imperfect CSIT.EURASIP Journal onWireless Communications and Networking 2013 2013:240.
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