Performance Evaluation of the V-BLAST CosetDetector

M. Magarini and A. SpalvieriDipartimento di Elettronica e Informazione, Politecnico di Milano

Piazza Leonardo da Vinci, 32, 20133 Milano, ItalyTelephone: +390223993450, Fax: +390223993413

Email: {magarini,spalvier}@elet.polimi.it

Abstract- In this paper we address the analytical performanceevaluation of the V-BLAST coset detector (CD). The V-BLAST-CD is a sub-optimal detector for spatial multiplexing MIMOsystems using non-binary constellations. In the V-BLAST-CDthe decision on the transmitted vector is taken by applyingthe maximum likelihood detector on a list of candidate vectorscontaining only a subset of the possible transmitted vectors.The list of candidate vectors is generated by applying orderedV-BLAST detection to the subsets induced by partitioning themultidimensional signal constellation according to the mappingby set partitioning principle. Noting that the detection processof the V-BLAST-CD is similar to that of list decoding of errorcorrecting codes, in order to evaluate the error performanceof the V-BLAST-CD we can adopt the same approach usedto evaluate the error performance of the generalized minimumdistance decoding. The accuracy of the analysis is demonstratedby comparing it to simulation results.

I. INTRODUCTION

Spatial multiplexing over a multiple-input multiple-output(MIMO) channel is a means to increase the spectral efficiencyof wireless communication systems [1]. We consider a schemewhere the input information sequence is de-multiplexed intonT sub-streams which are modulated by an M2-QAM mod-ulator and transmitted in parallel from nT antennas at thesame time and frequency. The detection of the transmitteddata symbols is performed by processing the signals receivedfrom nR > nT antennas. For such a scheme the performance,in terms of symbol error probability, is strongly dependent onthe technique that is implemented in the receiver to detect thenT transmitted sub-streams.The optimum detector, which is the maximum likelihood

detector (MLD), has a complexity proportional to M2nT. Dueto this exponential dependence, the complexity of the MLDcan be prohibitively large also for moderate values of Al andnT. The V-BLAST architecture, proposed in [1], is a sub-optimal detection technique that allows us to detect the nTtransmitted sub-streams while keeping the complexity low. Insuch a scheme, symbols are detected sequentially accordingto the well-known nulling and successive interference can-cellation process. The V-BLAST operations are equivalent tothose of a decision-feedback equalizer (DFE) that operatesat different stages in the spatial domain. Hence, the overallperformance may be limited by the first error probability ofthe spatial DFE. The performance loss can be mitigated if an

appropriate detection ordering is introduced. In [ ] the post-detection signal-to-noise ratio (SNR) is taken as the orderingcriterion. Despite its detection simplicity, the main drawbackof V-BLAST is that the diversity order in the early stagesis lower than in the next ones [2]. The diversity order of V-BLAST at the n-th processing step is nR - nT + n. This isthe main cause of the performance gap between V-BLAST andMLD (in the latter the diversity order is equal to nR).

Several sub-optimal detection strategies can be devised toreduce the performance gap compared to the MLD. In [3], forexample, the MLD is used to increase the diversity order offirst stages and only when the detected symbols are reliableenough can the detection be done with the V-BLAST algo-rithm. Note that for this detector the complexity is proportionalto M2ni, where ni represents the number of initial stagesto which the MLD is applied. The detector we focus on isthe V-BLAST coset detector (V-BLAST-CD) proposed in [4].The V-BLAST-CD is obtained by extending the principle ofreduced state sequence estimation [5], based on Mapping bySet Partitioning (MSP), to perform the detection in spatialmultiplexing MIMO systems using non-binary constellations.Its complexity is proportional to p32nT, where 1 < /_ < M2.In [4] it is shown that the V-BLAST-CD greatly outperformsthe V-BLAST detector at the cost of a slight increase ofcomplexity (t = 2 is considered). In particular, from low-to-intermediate SNR the performance of the V-BLAST-CDis the same as the MLD, while at high SNR the V-BLAST-CD still provides a significant performance gain over the V-BLAST detector. In [4] the benefits of the V-BLAST-CD aredemonstrated by computer simulations. In this paper we movea step toward a better understanding of the V-BLAST-CD. Tothis end, the performance evaluation carried out in this paperallows us to achieve further insights into the features of theV-BLAST-CD.The paper is organized as follows. The system model is

introduced in section II. A short description of the V-BLAST-CD is given in section III, while section IV addresses itsperformance analysis. In section V the results of the analysisare compared to simulation results.

II. SYSTEM MODELThe block diagram of the system that we consider is shown

in Fig. 1. Let a = [ai,a2,.. ,anT]T denote the vector of

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Fig. 1. Block diagram of the spatial multiplexing MIMO system.

transmitted symbols ((_)T denotes transposition). We assumethat symbols transmitted at a particular antenna are equallyprobable and take a value sm in the set S which containsall the M2-QAM constellation points. Let A denote the setof all M21TT possible equiprobable transmitted vectors. Thereceived signal vector at a particular time instant is representedin complex baseband form as

r = Ha+w,

where w = [Wi...., WnRlT is the noise vector of i.i.d.complex Gaussian random variables (RVs) with zero meanand variance o,2 and H is the nR x nT channel matrix whoseelements are i.i.d. RVs having uniform-distributed phase andRayleigh-distributed magnitude with average power equal to 1.H is independent of both a and w and it is assumed perfectlyknown to the receiver. The average radiated power from eachantenna is fixed to 1/nT. Thus, the total average radiatedpower is fixed to 1 and it tums out to be independent of thetotal number of transmitting antennas. The average SNR pertransmitted symbol at the receiver is =nR/(,nTO')

III. THE V-BLAST COSET DETECTOR

The principle behind MSP is a geometric partitioning ofthe signal constellation in subsets of diminishing size, in sucha way that minimum Euclidean distance within the subsetsincreases down the partition chain. A square M2-QAM con-stellation can be seen as a finite set of points carved out fromthe two-dimensional integer lattice 2. Similarly, the signaltransmitted from the nT antennas can be seen as belonging toa multidimensional constellation carved out from Z2nT. Thepartitioning of the constellation in subsets corresponds to thepartitioning of the lattice into a sublattice and its cosets. Inour proposal, the partition 72nT /2Z2nT is considered. Observethat, for each dimension of the M2-QAM signal constellationwe have the binary partition Z/2Z, that generates two sub-sets each containing M/2 points. The partition 72nT/2Z2nTdivides the multidimensional constellation into 22nT subsets,each containing nTM2/4 points. The ordered V-BLAST isused to perform detection for each of the 22nT subsets 1. Atthe end of this procedure a list of 22nT candidate vectors isgenerated. The decision is taken by applying the MLD to this

lNote that other sub-optimal detectors could be employed to perform thedetection in the subsets.

reduced set:=arg mim 11H~I2,a =argmin llr - Har l2

fir eAr

where ar is a vector taken from the reduced set Ar whoseelements are the 22nT candidate vectors. It is worth noting thatthe complexity of the MLD on the reduced set of candidatevectors is fixed and independent of the constellation size.

IV. PERFORMANCE ANALYSISPerformance evaluation of the V-BLAST-CD is carried out

by following the approach suggested in [6] to evaluate the errorperformance of the generalized minimum distance (GMD)decoding algorithm. We emphasize that the analysis of [6] canbe applied to any list-based sub-optimal detection scheme. Theerror probability of the M2-QAM symbol is upper boundedas

Ps (8) < Ps (£ILD) + P(Y), (1)where P(EAMLD) is the symbol error probability of the MLDover the set A and P(F) is the probability that the transmittedvector is not in the list of candidate vectors. The first term onthe RHS of (1) represents an upper bound to the conditionalsymbol error probability of the MLD in the reduced set ofcandidate vectors, where the condition is that the transmittedvector belongs to the reduced set A,. Note that when thepartitioning is such that we have just 1 constellation pointin each subset, as happens with a 4-QAM constellation, thenumber of candidate vectors is 22nT and P5(E) = P(eMLD).A. Derivation of PS(SMLD)

The term PS(EMLD) can be approximated as reported in [7],where a tight union bound and an asymptotic bound on thesymbol error probability are given for MIMO systems withtwo-dimensional signal constellations. The approximationson PS(eMLD) are obtained by calculating the symbol errorprobability corresponding to an arbitrary k-th (k = 1, . .. , nT)transmitted sub-stream. Let sm be the symbol transmitted bythe k-th antenna. We denote as Ai the subset of M2(nT-1)vectors in A which have the symbol sm in their k-th position.We also denote as Ai the set of (M2nT _ M2(nT-1)) vectorswhose k-th entry differs from Sm, The union bound onPS(EMLD) iS

Ps(EMLD) . M E E ( (2)SmnESjEAj iE"Ai

where PSm,,ij denotes the pairwise error probability betweenthe vectors ai E Ai and aj E Aj, given that sm is transmittedby the k-th antenna. As is shown in [7], the closed-formexpression of Ps ,ij is given by

PSm,J_ 1 fjl(2nRU- I n (3)

(1 + rsm,ij)2nR n- V n J l, (3

where

Ila, - aj 12 |ai-aj- I' + ai aj 112Sr,ij= + 454 2+ +±1w w ~~~~w

19

The computation of (2) would require up to M2nT (M2nTM2(nT-1)) pairwise error probability computations. However,a reduction of this number can be obtained if constellationsymmetries are exploited. At high SNR the Ps ,ij given in(3) can be expressed asymptotically as

PSmiis,j(nRwhere rsm,ij |a - aj 2W/.

B. Derivation of P(YF)As far as the term P(F) is concerned, the event F repre-

sents the probability that there is at least an error in the subsetcontaining the transmitted vector. An upper bound to P(Y)can be obtained by following the analysis derived in [2] forbinary modulations. The same approach is here extended tosquare QAM modulations by using the analysis presented in[8]. In what follows we refer to a generic QAM constellationwith M2 signal points. We point out that the probabilitieshereafter computed can be used as well when subsets of M2points are at hand, provided that the SNR is properly scaled.

According to [2] the upper bound on P(.F) isnT

P(F) < E Pe,n,n=l

(4)

where Pe,n denotes the average conditional symbol errorprobability at the n-th step of the detection process, wherethe condition is that no errors occur in the previous detectionsteps. The Pe,n is computed as [10]

Pe,n = Pe ()Pn(Y)dYI

where Pe(-y) is the symbol error probability of an M2-QAMsignal in an additive white Gaussian noise (AWGN) channelfor the instantaneous SNR -y and pn (-y) is the probabilitydensity function (pdf) of -y at the n-th detection step. Theprobability of error for an M2-QAM signal in the AWGNchannel can be expressed as [10]

P6(ay)= (M21) (4MQ (M IA )

obtain the expressions for the other cases but it involves moreelaborate mathematics. The pdfs of -y at the first and secondsteps for a 2 x 2 system are respectively

p1(y) = a e7/7c-2--y/_3 _ -yl-, (7)

(8)and

P2(Y) = 81 (1 + 2q )e--/ c,-2

where -/= n/fR denotes the average SNR per channel.Because the pdf given in (8) is derived by assuming thatthe symbol detected in the first step is correct, the Pe,2will represent the average symbol error probability at thesecond step given that no error occurs in the interferencecancellation step. Observe that in [2] this assumption is madein the derivation of the pdf of -y at each step of the V-BLAST detection process. It is important to note that the twoexpressions in (7) and (8) can be seen as a weighted sum ofchi-square pdfs with an even number of degrees of freedomand with different parameters. The expression of the pdf fora chi-square random variable with 21 degrees of freedom andaverage SNR per channel -yo is [10]

P(-Y) (lI

a1(1- -y>0. (9)

The closed form solution of (5) when (6) is averaged over (9)is given by

e(M2 DO, 1) = M2 (M-1+41 l(Al12I_yo,l)

-(M - 1)12(M2,-_oI 1)) I(l0)where the analytic expressions of Ih(M2,-yo, 1) andI2(M2, -yo, l) are given respectively by equation (9) andequation (10) in [8]. Note that the average symbol errorprobability given in (10) corresponds to that of an M2-QAMmodulation with 1-fold maximal ratio combining (MRC)space diversity in Rayleigh fading channels.By substituting (7) and (8) in (5) and using (10) the average

error probabilities at the first and second step are respectivelygiven by

Pe,,, = 2P (2, tY -4 Pe ( 72 7c a - e (M21 72c,2)46 2 42

(11)4(M- 1)Q2( M1)) (6)

whereQ(x)= i e- du.

The main contribution of [2] consists in the derivation ofthe pdf of -y at the different detection steps when the optimalordering is implemented by the V-BLAST detector. In [2]closed-form analytical expressions are given for the 2 x nRsystem, while tight bounds can be obtained for the generalnT x nR as shown in [9]. For the sake of simplicity, inthe following we derive the analytic expression of P6,n forn = 1, 2, in the 2 x 2 case. The same approach can be used to

Pe,2 P(M2 2) +±2 (M24 3). (12)

When (11) and (12) are used to compute the performance ofthe V-BLAST-CD described in section III, it is necessary totake into account that the number of constellation points ineach subset is M2/4, therefore the tY appearing in the aboveequations must be multiplied by 4(M2 - 1)/(M2 - 4).

V. EXPERIMENTAL RESULTSTo demonstrate the accuracy of the bound (1) we consider

a system with nT = nR = 2. Note that in this case thedetector of [3] could not provide any complexity reduction.The performance of the V-BLAST-CD is compared to that of

20

and

n = 1 1 ... I nT, (5)

100

10

1

c:Ul)

10

1n-5I10 15 20 25 30 35

average SNR (dB)Fig. 2. Performance comparison of V-BLAST, V-BLAST-CD and MLD fora 2 x 2 system with 16-QAM modulation.

the V-BLAST and MLD. Figures 2 and 3 report the simulatedSER (symbol error rate) versus the average SNR respectivelyfor 16-QAM and 64-QAM modulations. The SER is measuredby transmitting a sequence of 106 symbols from each antenna.Together with simulation results, the bounds on the symbolerror probability given in (l) for the considered detectiontechniques are also reported in the figures. We observe thatthe performance of the V-BLAST-CD algorithm is close tothat of the MLD for a SER greater than 10-3. In this regionthe bound in (1) is dominated by the error probability of theMLD on the reduced set of candidate vectors. For a lowerSER the performance of the V-BLAST-CD is dominated bythe symbol error probability of the ordered V-BLAST detectorin the subsets. As the figures show, the upper bound (1) is tightat SER lower than 10-3, where Ps(8) P(QF).

VI. CONCLUSION

The main contribution of this paper consists in the perfor-mance evaluation of the V-BLAST-CD proposed in [4]. Thisis a sub-optimal detector for non-binary modulation formatswhere the decision is taken by applying the MLD on a

reduced set of candidate vectors. The set of candidate vectorsis generated by applying the ordered V-BLAST detector tothe subsets obtained by a partitioning of the multidimensionaltransmitted signal constellation according to the MSP prin-ciple. By exploiting the conceptual similarity between thedetection process implemented by the V-BLAST-CD and thatof list decoding of error correcting codes, an upper bound tothe symbol error probability of the V-BLAST-CD has beenderived by following the same approach proposed in [6] forGMD decoding. The bound on the symbol error probability ofthe V-BLAST-CD is given by the sum of two terms. The firstone represents an upper bound on the symbol error probabilityof the MLD given that the transmitted vector is among the setof candidate vectors. The second one gives the probability thatthe transmitted vector is not in the list of candidate vectors.

10°

10-1

T- -2w 10

10-3 I

10 15 20 25 30 35 40average SNR (dB)

Fig. 3. Performance comparison of V-BLAST, V-BLAST-CD and MLD fora 2 x 2 system with 64-QAM modulation.

Noticeably, the error performance of the proposed detectorturns out to be close to that of MLD at low-to-intermediateSNR. Computer simulations have been used to demonstratethe accuracy of the analysis.

ACKNOWLEDGMENTThis work was supported by the Virtual Immersive Com-

munication (VICom) project of the Ministero dell'Istruzione,dell'Universita' e della Ricerca (MIUR).

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[2] S. Loyka and F. Gagnon, "Performance analysis of the V-BLASTalgorithm: an analytical approach," IEEE Trans. Wireless Commun., vol.3, pp. 1326-1337, July 2004.

[3] W. J. Choi, R. Negi, and J. M. Cioffi, "Combined ML and DFE decodingfor the V-BLAST system," in Proc. ICC, New Orleans, LA, pp. 1243-1248, 18-22 June 2000.

[4] M. Magarini and A. Spalvieri, "A suboptimal detection scheme forMIMO systems with non-binary constellations," in Proc. PIMRC,Barcelona, Spain, pp. 1579-1582, 5-8 Sept. 2004.

[5] M. V. Eyuboglu and S. U. H. Qureshi, "Reduced-state sequence estima-tion with set partitioning and decision feedback," IEEE Trans. Commun.,vol. 36, pp. 13-20, Jan. 1988.

[6] D. Agrawal and A. Vardy, "Generalized minimum-distance decoding inEuclidean space: performance analysis," IEEE Trans. Inform. Theory,vol. 46, pp. 60-83, Jan. 2000.

[7] X. Zhu and R. D. Murch, "Performance analysis of maximum likelihooddetection in a MIMO antenna system," IEEE Trans. Commun., vol. 50,pp. 187-191, Feb. 2002.

[8] C. J. Kim, Y. S. Kim, G. Y. Jeong, J. K. Mun, and H. J. Lee, "SERanalysis of QAM with space diversity in Rayleigh fading channels,"ETRI J., vol. 17, pp. 25-35, Jan. 1996.

[9] S. Loyka and F. Gagnon, "Analytical framework for outage and BERanalysis of the V-BLAST algorithm," in Proc. IZS, Zurich, Switzerland,pp. 120-123, Feb. 2004.

[10] J. G. Proakis "Digital communications (4th edition)," McGraw-Hill: NewYork, 2001.

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