SignaltoInterferenceRatiobasedAntennaSelection forSpatial...

Noname manuscript No.(will be inserted by the editor)

Signal to Interference Ratio based Antenna Selection

for Spatial Multiplexing

Subhendu Batabyal · Suvra Sekhar Das

Received: date / Accepted: date

Abstract To achieve reliable high throughput wireless links spatial multi-plexing and diversity modes of multiple-input multiple-output (MIMO) areused in combination. Antenna selection (AS) has minimal complexity amongother spatial diversity methods. Bell Labs Layered Space Time (BLAST) isa low complexity spatial multiplexing technique, especially when minimummean squared error (MMSE) receivers are considered. However, optimal ASis known to be computationally intensive when used along with spatial mul-tiplexing, since an orthogonal subset channel matrix is required to be found.Known suboptimal algorithms are still relatively complex and incur perfor-mance penalties. In this paper we propose a low complexity AS algorithm forBLAST. It uses an approximate signal to interference ratio metric as a heuris-tic measure to select a given number of antennas. It produces the selectionchoice after a single iteration only. A structure to reduce hardware complexityby reusing the MMSE equalizer block is also proposed. Such reuse can be ap-plied to several AS algorithms. We compare the performance of the proposedalgorithm against others using mean spectral efficiency (SE), 10% outage SE,symbol error rate performance and implementation complexity. The impact ofapproximate expressions used in the proposed algorithm is also analyzed.

Keywords Antenna Selection · Spatial Multiplexing · BLAST · MMSEReceiver · Resource Reuse · Computational Complexity

Subhendu BatabyalG. S. Sanyal School of TelecommunicationsIndian Institute of Technology, KharagpurTel.: +91-801-674-6110Fax: +91-322-228-2266E-mail: [email protected]

Suvra Sekhar DasG. S. Sanyal School of TelecommunicationsIndian Institute of Technology, KharagpurE-mail: [email protected]

2 Subhendu Batabyal, Suvra Sekhar Das

1 Introduction

Multiple-input multiple-output (MIMO) Antenna transmission significantlyenhances the spectral efficiency (SE) of modern wireless communication sys-tems by means of spatial multiplexing [1], [2]. Spatial diversity on the otherhand improves channel reliability [3], [4]. Spatial diversity can be achievedthrough several approaches, among which antenna selection (AS) involves oneof the lowest levels of transceiver complexity [5],[6]. Spatial multiplexing (SM)can be achieved through several approaches, including Bell Labs Layered SpaceTime (BLAST) [7], [8], singular value decomposition (SVD)–MIMO [9], [10]and codebook-based transmission [11]. Of these, BLAST with linear receiversis computationally inexpensive whereas SVD–MIMO involves significant com-putational complexity and very high feedback overhead, but with potentiallyhigher spectral efficiency [4], [12]. Both spatial diversity and spatial multiplex-ing gains are affected by imperfect channel estimates and branch correlation[13], [14].

A tradeoff between SM and some form of diversity, e.g. antenna selection[15], [16] would be desirable. The specific signal to noise ratio (SNR) and chan-nel correlation properties would determine the extent of use of each [16],[17].The diversity mode realized using AS provides lower gain compared to max-imal ratio combining (MRC). However, the transceiver architecture can besignificantly simplified with AS. Thus keeping implementation simplicity inmind, we focus on a combination of BLAST with minimum mean squared er-ror (MMSE) receivers and AS to achieve the best possible tradeoff betweenmultiplexing and diversity at a low cost.

Several works cover aspects of combining spatial multiplexing using linearreceivers and AS [16], [18], in order to attain the above-mentioned trade-off. Ofthese, some present the impact of imperfect channel estimates and correlatedbranches [19],[17], but few have been tested in presence of channel estimationerrors with real world outdoor channel models. Such channel models have beenproposed, among others, for 3rd Generation Partnership Project (3GPP) LongTerm Evolution (LTE) and LTE-Advanced (LTE-A)[20]. One such model isthe correlated Urban Micro-cell (UMi) channel model which is used for perfor-mance evaluation in this work. Transmit AS is shown to improve capacity overcorrelated MIMO channels [21],[22] and transmit as well as receive AS havebeen used to reduce transceiver costs [6],[23]. Most algorithms proposed for ASinvolve several iterations [22], [23]. They also involve computationally complexmatrix and/or vector operations like matrix inversion, singular value compu-tation, logarithm of determinants, post-processing SNR estimation [16], [24],[25] etc. An algorithm that does an exhaustive search over all possible combi-nations to select MTs

antennas from MT possible transmit antennas and MRs

antennas from MR possible receive antennas requires(

MT

MTs

)(

MR

MRs

)

iterations

((

AB

)

= A!B! (A−B)! , where x! indicates factorial x). Several attempts to reduce

the complexity of AS algorithms while retaining performance close to optimalhave been made [26],[27],[28]. Among the simplest of these are norm-based

Signal to Interference Ratio based Antenna Selection for Spatial Multiplexing 3

selection (NBS ) [29] and cross-correlation based selection (XBS ) [30]. NBShas significant performance penalties in correlated MIMO channels [26],[30].XBS has performance penalties that are smaller compared to NBS, but stillsignificant [30]. In this work we present an AS algorithm for use with SM,which uses a single iteration unlike the methods mentioned above. It usesan approximated signal to interference ratio (SIR) metric for antenna selec-tion procedure. The metric helps in reducing complexity of implementation.A reuse structure to further reduce hardware complexity of AS mechanism isalso presented. The detailed effect of using approximation is also provided inthe paper.

All of the algorithms mentioned above consider that the number of anten-nas to be selected is available as input. The number of antennas to be used fora particular connection may be determined by constraints such as the numberof available RF and modulator/demodulator chains [6]. Typically the num-ber of antennas is more than the number of RF chains. Studies regarding thechoice of the number of antennas typically take into account the channel cor-relation properties and possibly also its time variation [17], the constellationchosen for each transmit stream and corresponding error rates [16]. In caseof communication technologies, such as LTE-A, which uses multi-user MIMO(MU-MIMO) transmission [11], the particular configuration for a specific linkis selected by the Base Station (BS) such that the sum-user-rate is maximizedwhile maintaining fairness. These considerations are beyond the scope of thepresent work and hence we assume that the number of antennas to be selectedis available.

Typically these algorithms study the performance in terms of ShannonCapacity and relatively few of them use the modifications of Shannon Capacityformula, which account for the performance in wireless systems like LTE, ofpractical coding schemes, with signaling overheads [31]. In this work we analyzethe performance using the modified Shannon capacity formula and presentthe results for correlated MIMO channel as specified by ITU [20] consideringpractical operating SNR range of mobile communication systems. The meanand 10% outage capacity as well as symbol error rate performance for theproposed as well as other AS algorithms are presented.

The rest of the paper is organized as follows. Section 2 describes the sys-tem model. Section 3 describes the proposed AS algorithm. Sections 4 and 5compare the complexity and performance, respectively, of the proposed algo-rithm against existing algorithms used for reference. Section 5 also presents astudy of the performance penalties due to the approximations inherent in theproposed algorithm. Finally the conclusions are presented in Section 6.

2 System Model

Fig. 1 illustrates the system model considered in this paper. A typical spatialmultiplexing system comprising MT transmit antennas of which MTs

are se-lected and MR receive antennas of which MRs

are selected is shown. There is


a 1-1 mapping between the MTstransmit streams and the MTs

antennas se-lected for transmission (MTs

≤ MT). Stxu and Srx

u represent the set of selectedantenna indices at the transmitter and receiver, respectively. At the receiverthe ‘Training Mode Channel Estimation and Multimode Antenna Selection’module programmes the ‘Antenna Select Switch’ block to select MRs

out ofMR possible receive antennas (MTs

≤ MRs≤ MR). It also provides feedback to

programme the transmit side ‘Antenna Select Switch’ block. Quantities suchas MTs

, Stxu etc. are fed back as shown in the figure.

Pre

code

r (W

)

1

Rec

eive

Spa

ce−

time

eqP

roce

ssin

g (H

)

Con

stel

latio

n M

appe

r

Con

stel

latio

nD

emap

per

.

.

Surx

.. ..

.. .. ....

..

....

Su

tx

1 +

z1

+

z2

Mob

ile R

adio

Cha

nnel

2

BIT

SO

UR

CE

Bit

Sin

k

2 2nd selected antenna

Rejected AntennaxSelected Antenna

3rd antenna3

symbol description

MT

MTs

MRs

DE

MU

X1

to M

Ts

Ts

to 1

M

UX

M

Ts

Rs

out o

f MR

)(M

MR

1

2

3

Ant

enna

Sel

ect

Sw

itch

.

. .. .

. ..

Precoder Matrix Generation

Training Mode Channel Estimation,Multimode Antenna Selection, And

Constellation Size Descriptor

W

,.. ..

H

x

Bas

eban

d−R

F C

hain

Mat

rix

2

x

2

1TRANSMITTER (TX)RECEIVER (RX)

3

Ts

(Mou

t of M

)T

z

+

MRs

x

RsMM

Bas

eban

d−R

F C

hain

Mat

rix

Ant

enna

Sel

ect S

witc

h

Fig. 1 System Block Diagram for Spatial Multiplexing with Antenna Selection MIMOSystem

Considering ideal synchronization, a system with MT transmit and MR

receive antennas with flat Rayleigh faded channel between each pair of antennalinks, the baseband receive symbol vector is represented as

yMRx1 = HMRxMTWMT×MT

xMTx1 + zMRx1, (1)

where the subscripts indicate the matrix dimensions, x the transmit symbolvector,H is the matrix of channel coefficients,W is the precoder weight matrixand z is the vector of noise samples associated with the receiver in each of theMR receive branches.

Capacity of the above configuration without channel state information atthe transmitter (as in BLAST), whereW becomes identity matrix, while usingthe modified Shannon Formula [31], can be written as

C = βeff log2 det

(

IMRs+

ρ

MTs

HMRxMTsHH

MRxMTs

Υeff

)

MAX HMRxMTs⊂HMRxMT

, (2)


where det indicates determinant, βeff is the effective bandwidth and Υeff isscaling of SINR due to system parameters, average SNR ρ = P

σ2n

, P being

the maximum total power available and σ2n the noise variance of any receive

branch. The problem can now be stated as to find HMRxMTs⊂ HMRxMT

whichmaximizes capacity.

In real systems, the channel is estimated at the receiver. An element of theestimated channel matrix, denoted by HnR,nT

, is related to the correspondingelement of the actual channel matrix HnR,nT

, by:

HnR,nT= HnR,nT

+ enR,nT, (3)

where enR,nTindicates a complex Gaussian estimation error with zero mean

and variance σ2e = 1

2ρ per dimension, ρ being the average SNR at each receive

antenna. In this paper, we have used H (H) to loosely mean both the MRxMT

training-mode channel (estimated channel) matrix as well as the MRsxMTs

subset of the matrix which is active in data-mode and the use can be easilyidentified from the context of discussion. Time indices have been suppressedfor notational convenience.

The expression for x, the estimated transmitted symbol vector, at thereceiver after equalization, can be written as

x = Heqy, (4)

where the MIMO channel equalizer matrix Heq is specific to a transceiverconfiguration. Below we consider linear one stage MMSE receiver for BLAST,and derive its SINR expressions.

2.1 BLAST with MMSE receiver

The channel equalizer matrix for BLAST with single stage linear MMSE re-ceiver following [32] is,

Heq = ΠHH , given that, (5)

Π = [Rxx;σHHH + IMTs

]−1Rxx;σ, (6)

where (·)H indicates hermitian, (·)−1 indicates matrix inverse and IM is iden-tity matrix of size M ×M . Rxx;σ = ( 1

σ2n

)Rxx is the SNR dependent autocor-

relation matrix of the transmit vector x, σ2n being the noise variance of any

receive branch. The matrix Rxx = E[xxH ] = diag(P), where E[·] is the ex-pectation operator and P = [P1,P2, . . . ,PnT

, . . . ,PMTs] with PnT

denoting thepower transmitted on the nT

th antenna, and P is the maximum total poweravailable, so that

∑

nT∈StxuPnT

≤ P, Stxu being the set of selected (active)

transmit antennas. Combining (1), (4) and (5) we get

x = Π[HHHx+ HHz] = Π[q+ HHz], (7)


where q = HHHx represents the signal plus interference component of x

(apart from Π). The elements of the matrix L = HHH which is embedded

in the above expressions can be written as Lm,n =∑MRs

nR=1 H∗nR,mHnR,n, where

(·)∗ indicates the complex conjugate.The SINR for the nT

th stream, without the effect of Π can be expressedas

ΥnT=

|LnT,nT|2PnT

∑

k∈Stxu,k 6=nT

Pk|LnT,k|2 + σ2n

∑MRs

nR=1 |HnR,nT|2. (8)

The effect of Π (6) which multiplies both the terms within brackets in (7)is to smear the signal and interference terms embodied in q. Thus, the finalSINR expression ΥnT

is identical to the corresponding intermediate expressionΥnT

(8) but with J = ΠL in place of L, and F = ΠH in place of H.Since spatial waterfilling has a high computational complexity [1] involving

Θ(MT2) real multiplications apart from the SVD of H and further since it also

involves high feedback overhead while being sensitive to channel estimationerrors [17], we restrict the choice to equal power allocation over each transmitstream (= P

MTs

) in order to keep the complexity within limits. Thus,

ΥnT=

|JnT,nT|2

∑

k∈Stxu,k 6=nT

|JnT,k|2 +MTs

ρ

∑MRs

nR=1 |FnR,nT|2. (9)

In case of equal transmit power per antenna the matrix Π (6) simplifies to

Π = [HHH +MTs

ρIMTs

]−1. (10)

2.2 SVD–MIMO

In this work, we also include the analysis for SVD-MIMO, as it is known toprovide the highest possible throughput, for ensuring completeness in per-formance comparison. We consider the use of antenna selection as per themethods referred to in this work and the proposed antenna selection method.Equal power across all antennas is used for SVD-MIMO for parity as hasbeen considered for BLAST. Since we look at the capacity expressions, itcan be viewed in the light of per antenna rate control [33]. For SVD-MIMO,Heq = UH and W = V.

Therefore from (1) and (4) x = ∆x + UHz,where ∆ = UHUDVHV,H = UDVH being the SVD of H. If the channel estimates are exact, thenU = U, V = V, and ∆ = D, where D is the diagonal matrix consisting ofthe singular values of H. The SINR for the nTth stream can be expressed as

ΥnT=

|∆nT,nT|2PnT

∑

k∈Stxu,k 6=nT

Pk|∆nT,k|2 + σ2n

. (11)


Assuming, as in the case of BLAST with MMSE receiver, equal power overeach transmit stream (= P

MTs

) we have,

ΥnT=

|∆nT,nT|2

∑

k∈Stxu,k 6=nT

|∆nT,k|2 +MTs

ρ

. (12)

2.3 Capacity / Spectral Efficiency (SE) calculations

Denoting by CnT(H, ρ) the instantaneous SE of the nTth stream for a given

channel realization H and average SNR ρ, and by C(H, ρ) the correspondingtotal SE, we can write

C(H, ρ) =∑

nT∈Stxu

CnT(H, ρ). (13)

Representing the instantaneous SINR for the nTth stream as ΥnT(H, ρ), the

instantaneous SE for the corresponding stream can either be based on the orig-inal Shannon formula [34], or (more appropriately) modified Shannon Formulafor LTE[31]:

CnT(H, ρ) = βeff × log2

(

1 +ΥnT

(H, ρ)

Υeff(ρ)

)

. (14)

For comparing various AS algorithms, we study both the mean(

C = EH(C(H, ρ)))

and 10% outage SE [35] as functions of average SNR (ρ).

3 Proposed Antenna Selection Algorithm

The goal of optimal AS is to find the subset of transmit antennas (MTsout of

MT) and that of receive antennas (MRsout of MR) which yields the maximum

SE. Since per-stream SE is an increasing function of per-stream SINR, we usea selection procedure that searches for streams that have the highest SINR.Noting that the noise variance may be assumed to be equal for all branches,we propose to use the SIR instead of SINR in order to simplify the algorithm.Moreover instead of considering the signal and interference terms in the finalSINR expression ΥnT

(9) we prefer to compute them from the intermediateexpression ΥnT

(8) since the latter does not require a computationally expen-sive matrix left-divide operation while it captures the essential components ofSIR.

In the algorithm we use the estimate of L denoted by L = HHH to estimatethe SIR for the nT

th stream, (MTs= MT; here H is the estimated training

mode channel matrix). The SIR is:

ˆSIRnT=

|LnT,nT|2

∑MT

k=1,k 6=nT|LnT,k|2

. (15)


A justification for the selection of rows of L to compute per stream SIR canbe derived from two different sources:

1. The expression for L = HHH is found in the capacity expression for theMIMO channel [2]. This capacity is maximized when L is a scalar multipleof the identity matrix.

2. The algorithm for AS in [30] (XBS ) computes all the elements of L. Itfinds branch-pairs that have the highest mutual correlation and removesthe one that has a lower norm.

The proposed algorithm can be extended to receive AS by using Lrx = HHH

in place of L. A nice property of this algorithm is that it can be employed in awide variety of MIMO transmissions. It may also be noted that, with perfectchannel estimates, the eigenvalues of L are identical to the squared singularvalues of H.

The algorithm first computes SIR of each transmit antenna. Then it selectsthe MTs

antennas with the highest SIR. Computation of the per antenna SIRcan be done in one iteration, which is followed by ordering them. After removalof the undesired antenna(s), the H matrix is pruned to keep only the MTs

selected columns corresponding to the transmit antennas selected for datamode.

3.1 SIR based Antenna Selection (SBS)

The proposed algorithm for transmit AS (which is abbreviated as SBS, i.e.SIR based selection) is summarized in Table 1 below. The algorithm is to be

Table 1 Proposed (SBS) Fast Antenna Selection Algorithm

ALGORITHMSTEP-WISE

COMPLEXITY

txSelIdx = FastAntSel(MT, MTs, H)

L = HHH Ψ(MR,MT)for nT = 1 : MT

SIR(nT) =|LnT,nT

|2

∑MT

k=1,k 6=nT|LnT,k|

2

23MT

end

txSelIdx = SelMax(SIR,MT,MTs)

MTsmaxMT

J = SelMax(SIR,MT,MTs)

I = 1, 2, . . . ,MTfor k = 1 : MTs

J (k) = argmaxi∈I SIR(i)

I = I − J (k)end

used when MTs> 1 as NBS is optimal when MTs

= 1. The right column


of the table contains the computational complexity of significant steps in thealgorithm in terms of number of real multiplications and max/min selectionoperations. The numbers are placed side by side with the corresponding step.

In the proposed algorithm, after removal of the undesired antenna(s), the

H matrix is pruned to keep only the MTsselected columns corresponding to

the transmit antennas selected for data mode. Thereafter the expressions forper (selected) stream SIR change due to pruning, and this might result insuboptimal performance. The better option would be to reject one by one thetransmit stream which has the minimum SIR metric, and recompute the SIRmetric for the remaining streams after each rejection based on the pruned H

matrix. However, the complexity of this iterative rejection approach as perthe expressions for the no. of real multiplications (RMs - ΞSBS,iter

RM ) and real

additions (RAs - ΞSBS,iterRA ) in the Appendix (31, 32), is one order higher

compared to SBS. Furthermore we show in Section 5.4 that the gain withthis iterative rejection approach is relatively small even at high SNR for thespatially white channel. Therefore we choose not to consider the change inper stream SINR after pruning of the H matrix in the selection. Thus thecomputation of L and subsequently per-stream SIR (15) is only one-shot,involving a single iteration.

3.2 Reuse Structure

L = HHH (its complexity being given by the expression Ψ(MR,MT) (16, 20,21)), is a major contributor to complexity of SBS. For a MMSE receiver, thisoperation is part of equalization (6, 10). The corresponding block is used onlyin data mode and remains idle in training mode. Since the design of this blockneeds to account for worst case complexity (when all antennas are selected),it should be possible through its reuse to completely remove the complexity ofL computation. A reference diagram for such reuse is captured in Fig. 2. Thedashed line in Fig. 2 surrounds the reused portion of the MMSE equalizer. Thisportion computes the product of the input matrix and its conjugate transpose.In the training mode the input to this portion is the full MR x MT channelmatrix, while the output is switched to the input of the SIR computationblock. In the data mode the reused portion derives its input from the MRs

xMTs

selected channel matrix while its output is switched to the input of therest of the MMSE equalizer. Thus the dashed portion is reused for AS andMMSE equalization, leading to reduction in effective complexity.

3.3 Hybrid SBS Algorithm

Since NBS is expected to be the best algorithm when MTs= 1, the SBS

proposed above can be further modified. The first antenna may be selectedthrough NBS approach and the remaining antennas can be selected based onthe proposed metric. This algorithm is referred to as ‘Hybrid’ algorithm (HB)in the following discussions.


H

multiplymatrix

multiplymatrix

DemapperTo Constellation

Conpute SIRmetric for AS

conjugatetranspose

SIRChoose best

Training Mode Antenna Selection

Reused Block

Transition

Flow

Training Mode

Hsel

Data to Training Mode

Rest of MMSE Equalizer

Data to Training Mode transition

Data Mode FlowData (Y)

Fig. 2 Block Diagram for AS Complexity Reduction through Reuse of part of MMSEEqualizer for a MMSE receiver

4 Computational Complexity Analysis

The performance of the proposed algorithm is compared against some wellknown AS methods available in literature. The first being [26] which we referto as MaxCap as it is known to provide the highest SE. We also compare theperformance against [29] which we call as NBS and [30] which we name asXBS. NBS and XBS are known to be low complexity AS algorithms. NBSworks by selecting the antennas with the highest branch norm while XBS

selects the antennas with lowest mutual correlation.

The computational complexity of the proposed algorithm, MaxCap, NBSand XBS is presented in Table 2. Two types of operations are considered incomputational complexity analysis. The first being selecting Nsel maximum

or minimum values out of N real numbers (represented asNsel

max/min

N). Since

max/min selection complexity for all algorithms are similar and since it doesnot affect the overall hardware complexity much more attention is paid to-wards the second type namely arithmetic operations: real addition (RA) andmultiplications (RM). While computing arithmetic complexity the followingassumptions are made:

1. One complex multiplication (CM) involves 3 RMs and 5 RAs[36].2. A real division [37] and a square root [38] are both equivalent to a real

multiplication. AB

= A( 1B), and 1

Bcan be computed by table lookup.

It can be observed that MaxCap has the highest complexity and it increaseswith number of antennas. It is also observed that the proposed reuse structurecan reduce RM operations by up to a factor of 15 while it can reduce RAoperations by a factor of about 60. It can also be seen that with reuse XBS

has the lowest complexity, which is followed by SBS/ SBS-HB while NBS hasthe highest complexity if MaxCap is not considered.

Although XBS has the lowest complexity it has performance issues. If twobranches have a very high mutual correlation, then it will always drop thebranch with the lower norm since it must take decisions on pairs of branches


Table 2 Computational Complexity in Terms of No. of Real Multiplications (RMs) andAdditions (RAs), as well as Max/Min Selection of Various Fast AS Algorithms for 4x4 and8x8 MIMO systems. The reuse employed in case of some algorithms for effective complexityreduction is as shown in Fig. 2.

Algo

ComplexityReuse Max/Min Sel Arithmetic Operations

Employed? NxN 4x4 8x8(Y/N) Choose Nsel

RM RA RM RAY: Yes out of N

N: NoN

selmax/min

N

Max-N N

selmaxN

230 440 2478 5079Cap

SBSN

NselmaxN

136 192 928 1680

YN

selmaxN

32 8 128 48

HBN N

selmax

N128 190 912 1674

YN

selmaxN

24 6 112 42

XBSN

(N−Nsel)maxN

116 190 856 1660

Y(N−N

sel)maxN

12 6 56 28

NBS NN

selmaxN

32 28 128 120

at a time. This may degrade performance if the selected branch, unlike therejected one, has high correlation with the remaining branches too.

5 Results on Spectral Efficiency Performance

The performance of the proposed AS algorithm is evaluated in terms of meanand 10% outage SE vs. SNR considering a system having a maximum of 4transmit and 4 receive antennas (4x4). The results are presented for:

1. Shannon spectral efficiency in spatially uncorrelated channel [1], ideal chan-nel estimation hereafter referred to as Hw,

2. LTE spectral efficiency [31] in correlated UMi NLOS channel specifiedby International Telecommunication Union, Radiocommunication Sector(ITU-R) [20], with channel estimation error. Hereafter we refer to this sce-nario as Humin, where the ‘n’ in the subscript indicates ‘noisy’ channelestimates. We also refer to this as ‘realistic scenario’.

We focus our comparison of different algorithms on the average SNR range5-15 dB for mean SE and −6-0 dB for outage SE because in the UMi NLOSchannel, the downlink wideband SNR remains in the 0-15 dB range for 75%of the cases ([39], slide 41).

In transmit AS plots and tables, 1 < MTs< MT andMRs

= MR is assumed.MTs

= 1, for which NBS is optimal, and MTs= MT which does not involve

antenna selection are not considered. In order to reduce the number of plots,we have presented the results such that there is a switch between MTs

= 2 and


3 depending on whichever performs better (see Table 3) for the given scenarioand the average SNR (ρ) of operation.

Table 3 SNR Range for Selection of Best Performance Mode (1x4, 2x4, 3x4 or 4x4) withTransmit Antenna Selection in −6 : 21 dB SNR range

SM Mode, ChannelSNR-range for Selection(dB)(Criterion: mean SE)

1x4 2x4 3x4 4x4BLAST, Humin −6.0 : 6.0 6.0 : 21.0 – –SVD, Humin −6.0 : 4.4 4.4 : 11.0 11.0 : 20.5 > 20.5BLAST, Hw – −6.0 : 3.5 3.5 : 12.2 > 12.2SVD, Hw – −6.0 : −2.3 −2.3 : 6.2 > 6.2

5.1 Transmit Antenna Selection

In the performance plots for BLAST-TX-AS and SVD-TX-AS the performanceof the proposed algorithm is compared with MaxCap, NBS, and XBS. Sincethe performance of SBS and HB are identical, we mark ‘proposed’ against thecorresponding curve in the figures.

Fig. 3 shows the performance of transmit AS with BLAST in Hw. It isobserved from the figure that, the proposed scheme is better than NBS whichis better than XBS in both mean and 10% outage spectral efficiency. Themean SE of AS-BLAST is around 2.5% away from that of the optimal AS(MaxCap).

From Fig. 4 we get the SE performance for the SVD scheme describedearlier. The earlier observations hold good here as well. However, it is seenthat the performance of the proposed algorithm is indistinguishable from thatof MaxCap. This result further justifies the choice of the proposed metric forAS.

Fig. 5 shows the performance of transmit AS with BLAST in Humin sce-nario. It clearly shows the superiority of the proposed algorithm compared toNBS and XBS. The optimal AS (MaxCap) is ≈ 2% away in terms of mean SEfrom the proposed algorithm. The capacity gains achievable are ≈ 10%.

It is also seen that the proposed scheme has negligible performance gapwith optimal AS while being always better than other schemes when 10%outage capacity is compared.

Therefore it can be said that the proposed algorithm while having a com-putation complexity very close to XBS has significant SE gains over its com-petitors.


5 6 7 8 9 10 11 12 13 14 155

6

7

8

9

10

11

12

13

14

15

Average SNR (dB)

Me

an

Sp

ect

ral E

ffic

ien

cy (

bits

/s/H

z)

−4 −2 01

2

3

Average SNR (dB)

10

% O

uta

ge

SE

(b

/s/H

z)

MaxCap

Proposed

NBS

XBS

Fig. 3 Hw mean and (inset) 10% outage (Shannon) spectral efficiencies with BLAST-Transmit-AS using various AS techniques

5 10 156

8

10

12

14

Average SNR (dB)Me

an

Sp

ect

ral E

ffic

ien

cy (

bits

/s/H

z)

−6 −4 −2 0

1

2

3

Average SNR (dB)10

% O

uta

ge

SE

(b

/s/H

z)

MaxCap

Proposed

NBS

XBS

Fig. 4 Hw mean and (inset) 10% outage (Shannon) spectral efficiency with SVD-Transmit-AS using various AS techniques

5.2 Receive and Transmit Antenna Selection

Receive antenna selection always reduces mean SE vs. SNR performance -e.g. 3x3 performance is always poorer compared to 4x3 - and is employed


5 6 7 8 9 10 11 12 13 14 151.5

2

2.5

3

3.5

4

4.5

5

Average SNR (dB)

Me

an

Sp

ect

ral E

ffic

ien

cy (

bits

/s/H

z)

−4 −2 0

0.1

0.2

0.3

Average SNR (dB)

10

% O

uta

ge

SE

(b

/s/H

z)

MaxCap

Proposed

NBS

XBS

Fig. 5 Humin mean and (inset) 10% outage (LTE) spectral efficiency with BLAST-Transmit-AS using various AS techniques

to meet receiver cost and complexity constraints. The performance study oftransmit-plus-receive AS with the proposed and other AS algorithms yieldsresults that are very similar to those in case of transmit only AS. Hence, forthe sake of brevity, these results are omitted. It should suffice to say that thetwo-way pruning effect increases the gap between MaxCap and NBS, and SBS

as before lies in the intermediate region. Thus with transmit-plus-receive ASfor BLAST in Humin the mean SE loss relative to MaxCap and gain relativeto NBS at ρ = 15 dB is less than 9% and greater than 25% respectively. Interms of SNR margin the loss relative to MaxCap increases to ≈ 1.0 dB andthe gain relative to NBS increases to 2.7 dB.

5.3 Symbol Error Rate performance for BLAST-AS

The symbol error rate (SER) performance for QPSK in AS-BLAST for Hw

scenario is shown in Fig. 6. The Q() function [40] is used to map per-streamSINR into per-stream instantaneous SER. This is then averaged across streamsas well as realizations to get mean SER at the chosen SNR. The performanceof the proposed algorithm in terms of SER with a target SER of 10−5 is ≈ 1.8dB better than that of NBS and ≈ 4.2 dB better than that of XBS whilebeing almost identical to that of MaxCap. The diversity order for MaxCap(optimal) antenna selection is found to be close to the theoretical value of(MT − MTs

+ 1)(MR − MTs+ 1) = 9 [41]. Therefore it can be said that the


Fig. 6 SER for MIMO 2x4 SM with various AS algorithms and QPSK Transmission inSpatially White (Hw) Channel

proposed algorithm has SER performance matching the optimal AS.

5.4 Effect of Approximations in the Proposed Metric for AS

Since the proposed algorithm uses simplified expression to compute SIR (15),in this part we study the corresponding effects. We limit ourselves to BLASTAS 2x4 performance in spatially white channel with perfect channel estimates.The approximations used are:

1. The change of SIR upon pruning, i.e. removal of transmit branches byantenna rejection, has been ignored.

2. SIR has been used instead of SINR.3. Pre-multiplication by Π has been ignored.

Fig. 7 illustrates the incremental mean SE obtained by removing theseapproximations. The curve titled ‘SBS-iterative’ is generated by using theiterative rejection approach explained in Section 3. The benefit due to theiterative rejection which involves an order increase in complexity saturates inthe high SNR region at around .07 bits/s/Hz. This is because at high SNRthe MMSE equalizer coefficients (5), (10) tend towards the zero-forcing (ZF)equalizer coefficients [40] causing the effective channel matrix He = HeqH to


approach the identity matrix IMTs. The proposed metric loses its effectiveness

because there is minimal cross-correlation between branches. From (1) and (4)x ≈ x+Heqz and the only basis for AS in such a case can be the subset of Hleading to the lowest row norms of Heq which scales noise. At low SNRs, Π(10) ≈ ρ

MTs

IMTsand Heq (5) ≈ ρ

MTs

HH (similar to the matched filter) while

x (1), (4) ≈ ρMTs

Lx+Heqz. Thus the metric is more effective at lower SNRs.

Fig. 7 Mean SE losses due to approximations in the proposed SIR metric for antennaselection.

To find the SNR at which the proposed metric loses its effectiveness, we pro-ceed as follows. The effective channel matrix will become the identity matrixwhen, ∀i ∈ Stx

u , where Stxu is the set of all selected (active) transmit antennas,

the term ρMT

(∑MR

k=1 |hk,i|2) ≫ 1 =⇒ Y ≫ MT

ρwhere Y =

∑MR

k=1 |hk,i|2. Since

Y follows a gamma distribution, Pr(Y > y0) = e−y0(y3

0

3! +y2

0

2! +y0+1). Assuming

y0 = 10MT

ρ(since Y ≫ MT

ρ), and that Pr(Y > y0) = .995, we get ρ = 59.83,

i.e. 17.77 dB. We note that the incremental SE obtained by ‘SBS-iterative’in Fig. 7 saturates between 15 − 16 dB average SNR, which is close to thevalue obtained theoretically. Moreover, referring to Fig. 6 we observe that atvery high SNRs > 15 − 16 dB there is a noticeable difference between pro-posed and MaxCap SER curves. This further validates the obtained threshold.Since this threshold is above the typical region of operation of modern cellularcommunications, the metric is valid for most realistic cases.

The curve titled ‘SINR-COARSE’ is generated by using the SINR versionof the approximate SIR (8). As expected the benefit of using the noise term(on top of the interference terms) reduces at high SNR, when the noise is


negligible. At low SNRs the SIR metric becomes more and more applicablewhereas the denominator of (8) becomes dominated by the noise term whichdegrades performance because it no longer matches the SIR metric (15). Thisoutweighs the benefit due to noise term inclusion.

Next we observe that the ‘SINR’ curve, which represents the usage of (9)instead of (15), improves performance only at very low SNRs (≤ −2 dB). Athigh SNRs performance is worse than with (15). This is because at high SNRs

(9) becomes ΥnT≈

|JnT,nT|2

∑k∈Stx

u,k 6=nT

|JnT,k|2, where J ≈ IMTs

because of near-ZF

equalization. Since the denominator terms are very small compared to thenumerator for all streams, the selection can pick any random antenna subsetbased on small fluctuations. Although at first glance it may seem contraryto expectations, this degrades the performance, especially since pruning hasnot been considered. For ρ < −5.6 dB, ‘SINR’ outperforms ‘SBS-iterative’.Finally ‘SINR-iterative’, which uses the metric (9) but with a iterative rejectionapproach, gives benefit that also saturates at high SNR like ‘SBS-iterative’, forsimilar reasons. Its benefit exceeds that of the latter marginally for ρ > 11.7dB and ρ < −4.7 dB.

From the above discussion it is clear that the approximations used to arriveat the final SIR metric are valid ones. They have resulted in reduction ofalgorithmic complexity with maximum SE penalty ≈ only 1%.

6 Conclusion

We have introduced a low complexity, single iteration (one-shot) algorithmfor antenna selection based on SIR to be used in combination with spatialmultiplexing using MMSE receivers. The complexity as well as SE performanceof the proposed method are compared with those of MaxCap (maximum SE)and two low complexity AS algorithms - XBS and NBS. A reuse structure ofthe MMSE equalizer block is proposed. This helps in reducing complexity byan order of magnitude or higher. It can be applied to the proposed algorithmas well as XBS. Furthermore it is found that the complexity of the proposedalgorithm lies between XBS and NBS where XBS has the lowest complexitywith such reuse. Also, the complexity of the proposed algorithm with respectto MaxCap is an order of magnitude lower.

The SE performance of the proposed algorithm and its competitors is eval-uated in realistic ITU proposed channel conditions. The proposed low com-plexity algorithm is found to have SE which is ≈ 10% better than XBS, NBSwhile being roughly 2% away from optimal AS method. In spatially white sce-nario also, the proposed algorithm outperforms XBS and NBS. Therefore theproposed algorithm while having computational complexity close to minimumhas SE quite close to the optimal AS.

It has further been observed that the proposed algorithm has SER perfor-mance matching optimal AS while being 1.8 dB better relative to NBS and4.2 dB better relative to XBS.


We have also studied the effect of approximations used to arrive at theproposed SIR metric for AS. From this study we have shown that the approx-imations are valid and useful since they reduce algorithmic complexity withhardly any performance loss.

Therefore it can be said that the proposed algorithm is a good candidatefor AS to be used with SM given that it provides very low complexity withnear optimal SE performance.

Appendix

Note: Ξ represents complexity of the abbreviated algorithm mentioned in thesuperscript in terms of the number of operations of the specific type (RM/RAetc.) mentioned in the subscript. The algorithm MaxCap is abbreviated as‘MXCP’. Ψ and ΨA represent the multiplication and addition complexity (re-

spectively) of the operation L = HHH (see Table 1).

Ψ(p, q) =pq

6(3q + 1). (16)

ΨA(p, q) =1

4(7q2p− 3pq − 2q2). (17)

Part 1: Complexity Expressions Without Reuse

ΞMXCPRM = 3ΞMXCP

CM , where (18)

ΞMXCPCM =

2

3MTMR +

4

3+ 2MT

+max(MRs− 2, 0)(MT

2 +2

3MT +

1

3)

+(MRs− 1)Ψ(1,MT) + (

2

3MT + 1)(MR − 1)

+(MR +2

3)max(MRs

− 2, 0)(MR −MRs

+ 1

2).

ΞMXCPRA = 5ΞMXCP

CM + 2ΞMXCPRA1 , where (19)

ΞMXCPRA1 = 1 +max(MRs

− 2, 0)(1 + 2MT(MT − 1))

+(MR − 1)(MT + 1)

+MTmax(MRs− 2, 0)(2MR −MRs

− 1).

ΞSBSRM = 3ΞSBS

CM , where (20)

ΞSBSCM = Ψ(MR,MT) +

2

3MT

2. (21)

ΞSBSRA = ΨA(MR,MT) +MT(MT − 2). (22)


ΞHBRM = 3ΞHB

CM , where (23)

ΞHBCM = Ψ(MR,MT) +

2

3MT(MT − 1). (24)

ΞHBRA = ΨA(MR,MT) + (MT − 1)(MT − 2). (25)

ΞXBSRM = 3Ψ(MR,MT) +MT(MT − 1). (26)

ΞXBSRA = 2ΨA(MR,MT) +

MT

2(MT − 1). (27)

ΞNBSRM = 2MTMR. (28)

ΞNBSRA = MT(2MR − 1). (29)

Ω(p, q) =1

6[(p(p+ 1)(2p+ 1))− (q(q + 1)(2q + 1))]. (30)

ΞSBS,iterRM = (2 +

3MR

2)Ω(MT,MTs

)

+MR

4(MT −MTs

)(MT +MTs+ 1). (31)

ΞSBS,iterRA = MRΩ(MT,MTs

)

+(MT −MTs)(MT +MTs

+ 1). (32)

Part 2: Complexity Expressions With Reuse

Note: the underlined variables indicate complexity with reuse.

ΞSBSRM = 2MT

2. (33)

ΞSBSRA = MT(MT − 1). (34)

ΞHBRM = 2MT(MT − 1). (35)

ΞHBRA = (MT − 1)(MT − 2). (36)

ΞXBSRM = MT(MT − 1). (37)

ΞXBSRA =

MT

2(MT − 1). (38)

ΞMXCPRM =3(ΞMXCP

CM − (MRs− 1)Ψ(1,MT)−

2

3MTMR).

(39)

ΞMXCPRA =

5

3ΞMXCP

RM + 2ΞMXCPRA1 . (40)

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