Comparison of SVD-MIMO withAntenna–Selection–BLAST using Linear Receiversunder Channel Estimation Errors for ITU Channels

Suvra Sekhar Das1,2, Subhendu Batabyal11G.S.Sanyal School of Telecommunication, Indian Institute of Technology Kharagpur, India.

2Dept. of Electronics & Electrical Communication Engineering, Indian Institute of Technology Kharagpur, India

Abstract—In wireless communication systems, the spatial mul-tiplexing mode of Multiple Input Multiple Output (MIMO) isknown to provide significant gain in spectral efficiency in the highSignal to Noise Ratio (SNR) region. Between the two methods,Singular Value Decomposition (SVD) and Bell labs Layered SpaceTime (BLAST), for implementing spatial multiplexing, the SVDscheme provides much higher spectral efficiency but it is muchmore complex to implement than the BLAST scheme. The aimof this work is to present an algorithm for antenna selection,which works together with BLAST method and compare theperformance with that of SVD method. The comparison is doneconsidering linear receiver architecture, and that the receiver hasnoisy channel estimates. The channel model used for performanceevaluation is as per International Telecommunication Union(ITU)-specifications. It is shown in this work that, the proposedalgorithm used with BLAST is very simple to implement andperforms at par with SVD.

I. INTRODUCTION

Multiple Input Multiple Output (MIMO) antenna basedwireless communication systems provide the advantage oftransmitting several parallel data streams at the same time,in the same frequency (known as spatial multiplexing gain),thus increasing the spectral efficiency in terms of bits/s/Hz.The Singular Value Decomposition (SVD) based approachis known to achieve the highest capacity by orthogonalizingthe channel [1]. It uses either optimal water filling powerallocation or per stream rate control. Implementation of SVD–MIMO includes SVD operation which is computationallyintensive. It also needs a pre–coding matrix to be fed backwhich adds high amount of signaling overhead to the system.

In contrast, Bell labs Layered Space Time (BLAST) offersspatial multiplexing gain without requiring any feedback.BLAST with linear receivers is relatively simple to implement.However, the penalty paid by BLAST is in terms of capacity–loss. If complex non–linear receivers are used then capacity–loss can be reduced.

It was known that at low values of Signal to Interfer-ence plus Noise Ratio (SINR), diversity mode of operationis preferred to spatial multiplexing mode. However, severalworks have reported that a combination of spatial diversityand multiplexing get the best out of the channel [2]. Thediversity mode can be realized using simply the Alamoutischeme or using antenna selection or any other complex spacetime coding scheme. Antenna selection is one of the least

complex methods for achieving diversity gain. Therefore thefocus of this work is on a combination of antenna selectionwith BLAST which is expected to get both diversity and spatialmultiplexing gain while incurring the least additional receivercomplexity. The following discussion presents some existingwork on spatial multiplexing with antenna selection.

Several works combine antenna selection with spatial mul-tiplexing [3]–[7]. These works use Shannon capacity formulafor analysis and do not consider practical operating conditions,such as channel estimation error and International Telecom-munication Union (ITU) specified channel models for mobilecommunications. The reduction of capacity due to correlatedchannel is shown in [8]. It is shown that capacity can beimproved by using dynamic stream selection [9], [10]. It hasbeen shown, based on system design considerations takinginto account coded error performance with Hybrid ARQ (H-ARQ), that spatial multiplexing outperforms diversity in op-erating SNR conditions of modern wireless communications,specifically Long Term Evolution (LTE) [11]

It is well known that SVD-MIMO can achieve the high-est capacity amongst all other spatial multiplexing methods.However, the performance results of such schemes in practicalchannel conditions, which is given in the ITU channel models,as well as considering channel estimation error and includingthe performance of the physical layer of broadband wirelesscommunication systems such as the 3GPP-LTE is not wellknown. Some relevant papers are [12] and [13].

With the above background, the aim of this work is toshow that, by using a very simple suboptimal method ofantenna selection along with BLAST scheme, capacity whichis very close to SVD can be achieved under practical operatingconditions of mobile communication systems. The method forantenna selection described in this paper is computationallysimple and achieves a performance which is very close tothat of SVD. The performance is evaluated in ITU specifiedchannel conditions [14] using modified Shannon capacity [15],which represents the performance of the Orthogonal FrequencyDivision Multiple Access (OFDMA) physical layer of 3rdGeneration Partnership Project (3GPP)–Long Term Evolution–Advanced (LTE-A). LTE-A specifications include Turbo codesfor Forward Error Correction (FEC) and H-ARQ.

The rest of the paper is organized as follows. Section II

978-1-4244-8327-3/11/$26.00 ©2011 IEEE

describes the system model. The performance analysis ofthe different schemes is presented in Section III. Finally theconclusions are presented in Section IV.

II. SYSTEM MODEL

The framework for analyzing BLAST is presented below.MIMO transmission for flat fading is represented in basebandas

Y = HX + N, (1)

where X, which is a Ntxx1 vector, represents the Transmitdata symbols over the Ntx antennas, Y represents the Nrxx1symbol vector to be decoded at the receiver, N is the Nrxx1vector of noise samples associated with the receiver in eachof the Nrx receive branches and H is the matrix of channelcoefficients of dimension NrxxNtx. The vector X consists ofNtx independent Quadrature Amplitude Modulation (QAM)transmit symbols Xntx , where ntx is the transmit antennaindex. An element of the channel matrix, denoted by Hnrx,ntx ,is the channel coefficient for the nrx

th receive antenna andntx

th transmit antenna. Throughout this paper, time indiceshave been suppressed for notational convenience.

The expression for X, the Ntxx1 estimated transmittedsymbol vector, at the receiver after equalization, can be writtenas

X = HeqY, (2)

where the NtxxNrx MIMO channel equalizer matrix Heq isspecific to a receiver configuration. Since we consider linearone stage Minimum Mean Square Error (MMSE) receiver,therefore

Heq = UHH , given that (3)

U = [HHRxxH + Rnn]−1, (4)

where (·)H indicates hermitian, (·)−1 indicates matrix inverse,H is the estimated channel matrix (assuming least squareschannel estimation), Rnn is the noise variance matrix whichis σ2

nINtx , IM is identity matrix of size M × M and σ2n

is the noise variance of any receive branch. The matrixRxx = E[XXH ] = diag(P) where, E[·] is the expectationoperator, P = [P1,P2, . . . ,Pntx , . . . ,PNtx ], Pntx being thepower associated with the ntx

th transmit antenna branch andthe corresponding transmitted symbol.

Combining (1), (2) and (3) we get

X = U[HHHX + HHN]. (5)

The second component of (5) represents the processed noiseterm while the first component represents the desired symbolsand interference. The first component of (5) contains the termL = HHH whose elements can be written as

Lm,n =Nrx∑

nrx=1

H∗nrx,m × Hnrx,n, (6)

where (·)∗ indicates the complex conjugate. The term Q =HHHX in (5) represents the signal plus interference compo-nent of X (apart from U). The ntx

th element of this vector(corresponding to the ntx

th detected stream) is:

Qntx =∑k∈S

Lntx,kxntx

√Pk (7)

= Lntx,ntxxntx

√Pntx +

∑k∈S,k �=ntx

Lntx,kxntx

√Pk,(8)

where S = {1, 2, . . . ,Ntx} and k, an element of S, is anantenna index.

The term Lntx,ntxxntx

√Pntx in (8) is the desired sig-

nal component, while the summation over terms for whichk �= ntx is the interference from other transmitted streams.

Therefore the SINR for the ntxth stream, ignoring the effect

of U can be expressed as

Υntx=|Lntx,ntx |2σ2

xPntx

σ2x

∑k∈S,k �=ntx

Pk|Lntx,k|2 + σ2n

∑Nrxnrx=1 |Hnrx,ntx |2

, (9)

where, σ2x = 1, and P is the maximum total power available,

so that: ∑ntx∈S

Pntx ≤ P. (10)

It is seen from (9) that the symbols of the other streamsinterfere with the desired stream. Therefore in BLAST inter-stream interference is a critical factor.

However, the effect of U (4) is to smear the signal andinterference terms in L, as a result of which J = UL takes theplace of L in the final SINR expression. The noise scaling alsochanges in the final SINR expression. The scaling of σ2

n is withthe magnitude squares of the elements of the matrix D = UHinstead of those of H. Thus the final SINR expression standsas follows:

Υntx=|Jntx,ntx |2σ2

xPntx

σ2x

∑k∈S,k �=ntx

Pk|Jntx,k|2 + σ2n

∑Nrxnrx=1 |Dnrx,ntx |2

. (11)

The aim of the communication system is to maximize the totalsystem capacity

CTot =∑

ntx∈S

Cntx , (12)

where the Cntx represents the capacity (bits/s) of the ntxth

transmit stream / antenna as per the modified Shannon for-mula [15], which represents the throughput (bits/s) of thePHYSICAL layer of LTE-A,

Cntx = BwEff × log2

(1 +

Υntx

ΥEff

). (13)

This optimization is to be performed while satisfying theconstraints (10), and

Perntx≤ Pero∀ntx ∈ S, (14)

where Pero is the maximum tolerable Packet Error Rate (PER),and Perntx

is the PER for the ntxth stream. The objective can

be attained by adapting,

Pntx ,bL[ntx], for ntx ∈ S, (15)

where bL[ntx] is the bit load for the ntxth transmit stream

for a single instance of channel use (0 ≤ bL[ntx] ≤ bmaxL )

and Pntx ≥ 0. However, (13) encapsulates the Pero constraintas in [15]. In the current work dynamic per stream poweradaptation is not considered. Therefore

Pntx =P

Nseltx

,∀ ntx ∈ Su, (16)

= 0 otherwise, (17)

where Su is the set defined as:

Su = {ntx|Pntx �= 0} (18)

and Nseltx = Su is the number of such streams with non-

zero transmit power. From the above conditions, Pntx =0, or bL[ntx] = 0 implies that there is no transmission fromthe ntx

th transmit antenna in which case Nseltx < Ntx. When

the number of active transmission streams is less than Ntx,transmit power per antenna / stream will increase and viceversa in accordance with (10).

The SVD approach to maximize per-stream SINR is toorthogonalize the channel leading to a negligible inter-streaminterference in the absence of channel estimation error. AsSVD is undoubtedly the spatial multiplexing scheme with thebest performance when channel estimation accuracy is high,we consider its performance as the reference. The objectivesof the proposed scheme are to

1) be computationally less complex,2) involve less feedback overhead, and3) perform reasonably well even under channel estimation

error for practical ITU channels, where the MIMOchannels are often found to be correlated.

Observing (11) closely, one can find that Υntx decreaseswith increasing interference from other streams. To improveΥntx , we can consider reducing Nsel

tx , which will reduce thenumber of interference terms in (11). However, reducing Nsel

tx

will in turn increase Pntx as P is constant (16), and thiswill lead to an increase in signal as well as interferencepower. Considering the extreme cases of Nsel

tx = 1 at one endwhere there is zero cross-stream interference and no spatialmultiplexing gain, and at the other end Nsel

tx = Ntx, wherethere is a high amount of cross-stream interference as well asthe potential for a high spatial multiplexing gain, we foreseethat the optimum capacity for a given SNR will involve acombination of spatial multiplexing and diversity. From a pointof view of simplicity, the best choices for spatial multiplexingand diversity are BLAST and Antenna Selection, respectively.Thus we propose BLAST with Antenna Selection.

Antenna selection can be done, among others, with one ofthe following simple strategies:

1) Reduce both the number of transmit and receive anten-nas, keeping them equal

• Choose the sub-set of streams which will maxi-mize the sum capacity, simultaneously rejecting thestreams with the lowest SINR

2) Reduce the number of transmit streams through themethod of selection as in step 1 and use all receiveantennas.

In the latter method, an ideal transmit antenna selectionalgorithm would have to select one out of

∑k∈S

(Ntxk

)=

2Ntx−1 possible transmit antenna configurations. In the formermethod, the number of choices is even higher (

∑k∈S

(Ntxk

)2).

To avoid the exponential number of choices with the exhaus-tive search, we propose a single iteration algorithm that willcompute a reliability metric for each stream and then selectNsel

tx streams with the maximum reliability metrics. Sincecomputing the SINR as per (11) is complex, the Signal toInterference Ratio (SIR) of L = HHH as in (9) (ignoring theeffect of U and noise) is proposed as a simpler metric to beused for selecting the best streams. Denoting the estimate ofL by L = HHH, the estimated SIR is for the ntx

th stream is

SIRntx =|Lntx,ntx |2Pntx∑Ntx

k=1,k �=ntxPk|Lntx,k|2

. (19)

After computing (19) for each stream, we select the Nseltx

streams with maximum SIRntx . We analyze the performancefor Nsel

tx = 2, and 3. In contrast to the earlier works whereall possible combinations are used to select the stream withthe highest channel capacity, in this work only one iterationis being proposed. Further, the SIR is calculated from HHHand not post processing SINR. Calculation of post processingSINR requires two additional matrix left divide operations(J = UL, and D = UH) with Θ(nlog2 7) complexity [16].

III. ANALYSIS AND DISCUSSION

The performance evaluation is done using the modifiedShannon capacity as described above. This formula repre-sents the results for physical layer associated with LTE-A. Otherwise detailed physical layer simulation needs to beimplemented. LTE Physical layer specifications include rateadaptation with a large modulation and coding set (MCS)including QPSK, 16-QAM and 64-QAM modulation andFEC (Turbo coding with rates varying from 1

6 to 45 ). The

maximum allowable PER is 10%. To compute (13), the Υntx

is calculated from post processing SINR considering MMSElinear receiver for BLAST with or without antenna selectionas in (11). For Receive diversity (Rx-div) and Space TimeBlock Code antenna selection (STBC AS) modes, the MaximalRatio Combining (MRC) receiver is considered, and the postprocessing SINR computed accordingly. For SVD mode, thepost processing SINR is computed based on the standardunitary equalizing matrix based receiver.

The proposed scheme and algorithm is evaluated consider-ing a system having a maximum of 4 transmit and 4 receive

antennas (4x4). The results are presented for Urban Micro–cell (UMi) Non Line Of Sight (NLOS) channel specified byInternational Telecommunication Union, RadiocommunicationSector (ITU-R) [14] with channel estimation error, and a briefmention is made of the results for uncorrelated channel withperfect channel estimates. Channel estimation in the presenceof noise using Least Squares (LS) method is considered [1].The mean capacity and the 10% outage capacity are presentedfor comparison.

In the figures AS indicates Antenna Selection (AntSel),SM indicates Spatial multiplexing with BLAST using MMSEreceiver, while SVD indicates SVD-MIMO mode. NxM in-dicates an N transmit & M receive antenna system obtainedby selecting from a 4 x 4 system, except for Rx-div 1x4,where only 1 transmit antenna is present. Antenna selection isimplemented using the simple suboptimal algorithm describedin this paper using the metric as in (19), except for STBC 2x4where highest column norm of the estimated channel matrix isused for selecting the transmit antenna branches. For antennaselection with SVD (AS-SMSVD2x2 and AS-SMSVD3x3),instead of implementing antenna subset selection based on anexhaustive search for the best singular values of H, we haveused the same metric as in (19). This is because

• An exhaustive search for the antenna subset that max-imizes SVD-MIMO capacity has a high computationalcomplexity, and suboptimal approaches must be used.

• The benchmarking of the proposed suboptimal algorithmfor antenna selection with BLAST (as per 19) is notaffected by such a choice.

Most of the results for the Shannon Capacity of variousMIMO modes in uncorrelated channel with perfect channelestimates match the theoretical values presented in variousliterature, for eg [1] and are omitted due to space limitation.

A. LTE-A Capacity with true Channel Estimation for UMichannel as per IMT-A

This section presents results (mean and 10% outage ca-pacity) with true Channel Estimation for ITU specified UMichannel as per IMT-A through Fig. 1 and Fig. 2. Results withother channels are not presented for space limitations.

It is seen that 2x4 AntSel using the proposed simple methodof transmit antenna selection along with BLAST has thehighest mean capacity up to (Average) SINR of 7 dB andoutage capacity within .1 bps/Hz of the maximum outagecapacity (with SVD MIMO, and closely followed by STBC2x4 for the specified SINR range) up to SINR of 5 dB. Themean capacity of 2X4 AntSel is within .05 bps/Hz of thatSVD MIMO even at 9 dB SINR. It is important to note thatthe outage SINR range in modern cellular system is betweenSINR of -5 dB to 3 dB.

A key point to note here is that spatial multiplexing withor without diversity is optimal even at low SINRs. However,SVD MIMO requires the feedback of the estimated channelfrom the receiver to the transmitter, especially for a FrequencyDivision Duplex (FDD) system where the uplink and downlinkchannels are independent. This causes a significant penalty in

−5 0 5 10 15 20

1

2

3

4

5

6

7

8

Average SINR (dB)

Mean Spectral Efficiency (bits/sec/Hz)

MMSE SM 4x4SVD SM 4x4AS SM 3x3AS SM 2x2Rx−div 1x4STBC AS2x4AS−SMSVD2x2AS−SMSVD3x3AS−SM 2x4AS−SM 3x4

Fig. 1: Comparison of Mean LTE-Physical Layer Capacity VsSINR for different MIMO Schemes with true Channel Estimationfor UMi NLOS channel as per IMT-A

−5 0 5 10 15 20

1

2

3

4

5

6

Average SINR (dB)

10% Outage Spectral Efficiency (bits/s/Hz)

MMSE SM 4x4SVD SM 4x4AS SM 3x3AS SM 2x2Rx−div 1x4STBC AS2x4AS−SMSVD2x2AS−SMSVD3x3AS−SM 2x4AS−SM 3x4

Fig. 2: Comparison of 10 % outage LTE-Physical layer CapacityVs SINR for different MIMO Schemes true Channel Estimationfor UMi NLOS channel as per IMT-A

terms of capacity for all SINRs, as is obvious from Fig. 3 andFig. 4.

For the capacity plots with feedback overhead we haveconsidered the following

1) Antenna selection requires the feedback of the rejectedantenna index, requiring log2 Ntx bits for every rejected(transmit) antenna.

2) SVD MIMO requires the feedback of Ntx×Nrx complexcoefficients, each quantized to 8 bits for both real andimaginary components.

3) Rate 12 encoding is assumed for all feedback parameters.

Each of the above items must be updated every 1ms TransmitTime Interval (TTI).

IV. CONCLUSION

A low complexity algorithm for antenna selection to beused in combination with BLAST which achieves a spectral

−5 0 5 10 15 200

1

2

3

4

5

6

7

Average SINR (dB)

Mea

n S

pect

ral E

ffici

ency

(bi

ts/s

ec/H

z)

MMSE SM 4x4SVD SM 4x4AS SM 3x3AS SM 2x2Rx−div 1x4STBC AS2x4AS−SMSVD2x2AS−SMSVD3x3AS−SM 2x4AS−SM 3x4

Fig. 3: Comparison of Mean LTE-Physical Layer Capacity VsSINR for different MIMO Schemes with true Channel Estimationconsidering capacity loss due to signaling overhead for UMiNLOS channel as per IMT-A

−5 0 5 10 15 200

1

2

3

4

5

Average SINR (dB)

10%

Out

age

Spe

ctra

l Effi

cien

cy (

bits

/sec

/Hz)

MMSE SM 4x4SVD SM 4x4AS SM 3x3AS SM 2x2Rx−div 1x4STBC AS2x4AS−SMSVD2x2AS−SMSVD3x3AS−SM 2x4AS−SM 3x4

Fig. 4: Comparison of 10 % outage LTE-Physical layer CapacityVs SINR for different MIMO Schemes true Channel Estimationconsidering capacity loss due to signaling overhead for UMiNLOS channel as per IMT-A

efficiency very close to that of SVD is presented in this work.The performance is analyzed for linear receivers consideringchannel estimation error in ITU specified channel modelsusing capacity abstraction for LTE. The results (without con-sidering the signaling overhead) indicate that the 10% outagecapacity of the scheme is at par with the best schemes forup to 5 dB SINR, and the mean capacity is comparablewith SVD up to SNR of 9 dB. It can be said that theproposed method may be chosen by the Base Station basedon the average SNR condition up to 9 dB of SNR. Sincein broadband wireless mobile communication systems, up to70% of the users experience SINR below 9dB, the proposedmethod is expected to provide significant benefits in terms ofhigh capacity at reduced complexity. If signaling overhead isconsidered then the proposed scheme is preferred throughout

the operating region of SNR which is -5 to 20 dB.

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