Linear Signal Detector for V-BLAST Scheme Basedon Probabilistic Data Association

Lingyang Song, Student Member, IEEE, Are Hjørungnes, Senior Member, IEEE andHaopeng Li Member, IEEE,.

Abstract— In this paper, we propose a linear probabilisticdata association (PDA) based signal detector for V-BLAST. Themain idea is to utilize the principle of a pre-whitening filter torealize single symbol detection, similar to virtual single inputand single output (SISO) system, by removing the effects ofpast decisions. Simulation results show that our scheme canprovide performance very close to ML decoding with extremelylow computational complexity.

Index Terms— V-BLAST, MMSE, maximum likelihood detec-tion, Probabilistic Data Association.

I. INTRODUCTION

THE substantial advantages available in wireless systemscan be realized by using multiple receive/transmit anten-

nas (known as ”multiple-in, multiple-out” or MIMO channels).This has led to the development of Lucent’s ”Bell-Labs Lay-ered Space-Time Architecture” (BLAST) [1] [2] to increasethe transmission rate linear in the minimum number of theantennas at transmitter or receiver side. From communicationcapacity point view, V-BLAST [2] aims to maximize spatialmultiplexing gain, and it thus can be able to realize tremendousbandwidth efficiencies and then enhance system capacity.Many algorithms in the literatures can be applied to recoverthe signals in V-BLAST schemes, such as maximum likeli-hood detection (ML), minimum mean square error detection(MMSE) [5]. ML is optimal from the performance point ofview, but it requires prohibitive computational complexity, andMMSE is less complex but it leads to great performance lossas it does not consider the covariance of the noise. In [2], ansimple interference cancelation detector (IC) was proposed,but it also causes performance loss since it treat un-detectedsymbols associated with their channel coefficients as noise.Sphere decoder (SD) tend to perform very well [3], but sufferfrom the fact that their complexity is a random variable whichdepends on the channel realization [4].

Recently, some detection algorithms have been reportedbased on probabilistic data association (PDA) [6]- [8]. How-ever, [6] is proposed for space-time block codes (STBC)[9]; [7] and [8] can provide performance converged to MLdetector, but [7] aims to solve multiuser detection problemsfor CDMA system and in [8], it is mainly designed for

This work was supported by the Research Council of Norway (NFR),project number: 176773/S10.

Lingyang Song and Are Hjøungnes are with UniK-University GraduateCenter, University of Oslo, N 2007, Norway (e-mail: lingyang@unik.no;arehj@unik.no).

Haopeng Li is now with the IT Center, China Petroleum Engineering andConstruction Corp, Beijing, China (e-mail:lihaopeng@cpecc.com.cn).

large systems and its complexity also highly relies on theparticular channel conditions which can be very unstable likeSD algorithm. Hence, on the basis of the work in [2]- [8], wepropose a linear signal detector (LSD) using probabilistic dataassociation for V-BLAST schemes. The main idea is to utilizethe principle of a pre-whitening filter to realize single symboldetection, similar to virtual single input and single output(SISO) system, by removing the effects of past decisions.Both analytical and simulation results show that our schemecan provide performance very close to ML decoding withextremely low computational complexity. Closed-form SERis obtained, which can be also treated as a close upper boundfor V-BLAST scheme which does not seem to be done inthe previous literature. Simulation results show that there onlyexists a small gap between LSD and ML decoder.

Note that there are several other features of the LSD. Unlike[2], we consider the covariance of the undetected terms and thenoise. Unlike [7] which need to update the mean and varianceof each detected term, decision feedback process is employedto remove the effects of the already detected symbols. Besides,our detector only has one decision trial in comparison to thedetector in [7] that requires a certain number of iterationsto make the performance converged, and the one in [8] thathas to select many decision paths in order to realize optimalperformance. Another big difference compared to [6] is thatit requires to consider very complicated covariance but LSDnot. And also we provide analytical results and realize a virtualSISO for V-BLAST scheme that can cast some lights on thework in the transmitter end.

The rest of the paper is organized as follows. In SectionII, we introduce the system model. In Section III, we describethe proposed linear LSD based detector. IV. Simulation resultsare shown in section V. In section VI, our main conclusionsare made.

II. SYSTEM MODEL FOR V-BLAST SCHEME

Without the loss of generality, assuming a wireless systemoperating with M transmit antennas and N receive antennas,the received signal can be then expressed as:

r = Hs + n (1)

where r = [r1· · ·rN ]T with dimension N×1, H has dimensionN × M , whose element is complex Gaussian circularly dis-tributed with independent components all having unit varianceand zero mean, s = [s1· · ·sM ]T with size M × 1, n =[n1· · ·nN ]T having size N ×1. The noise variable nj consists

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of independent samples of a zero-mean complex Gaussianrandom variable with variance σ2 = E/(2SNR). E denotestotal energy that is normalized to M .

Although, the ML decoding can still be employed, it re-quires the exhaustive searching of all the possible combina-tions of the transmitted symbols over the known channel. Inthe following, we will deal with this issue by means of pre-whitening detector.

III. PDA BASED SIGNAL DETECTOR FOR V-BLASTSCHEME

A. PDA based Signal Detector

In order to apply LSD to V-BLAST, power ordering (PO)is required, which suggest that he signals that have relativelylarger channel power should be decoded before the signalswith smaller power. Suppose that ‖h1‖2 ≥ ‖h2‖2 ≥ ... ≥‖hN‖2, where hi represents the i-th column of H. Suppose sk

is the target and the previous k−1 symbols have been detected,the received signals in (1) can be further rewritten into threeparts, the past decisions, the target, and the un-detected terms:

r =k−1∑j=1

sjhj + skhk +M∑

j=k+1

sjhj + n (2)

where hi stands for the i-th column of H with dimensionP × 1. The pre-whitening detection can be performed in thefollowing:

1) Past Transmission Removal: Reconstruct the previous

effects according tok−1∑j=1

sjhj , and remove them from the total

received signals:

rk = r −k−1∑j=1

sjhj = skhk +M∑

j=k+1

sjhj + n (3)

2) Gaussian Approximation: One feature in this paper isthat we consider the joint effect of the un-detected terms andthe noise together by using Gaussian approximation. Hence,in (3), we assume that:

rk = skhk + ηk (4)

where η is a zero-mean complex Gaussian random variablewith variance Λk =

(Hk+1,M HH

k+1,M

) |s|2 + σ2IM withdimension M × M , |s|2 = 1 representing the average powerof the symbols in constellation A, IM denotes M×M identitymatrix, Hk+1,M = [hk+1...hM ] with size M × (M − k) and(·)H stands for conjugate transpose.

3) Pre-Whitening, and Matched Filter: The major con-tribution in this paper is that we transform the correlatedMIMO into a virtual SISO by Gaussian approximation andpre-whitening filtering. Note that Λk is a positive-defined Her-mitian matrix and linear transform will preserve the propertiesof the original system. After applying a pre-whitening filter,Θk = Λ−1/2

k , to both sides of (4), we get:

Θkrk = skΘkhk + Θkηk (5)

where the updated noise term Θkη consists of un-correlatedGaussian variables with zero mean and unit variance. Since the

above transformation is linear, the optimal performance of (4)will be preserved. Since the noise terms are un-correlated, amatched filter, Φ = (Θkhk)H , can be further employed reachthe following scalar form:

yk = ρksk + νk (6)

where yk = (Θkhk)HΘkrk, ρk = ‖Θkhk‖2, ‖x‖2 = xHx,and νk = (Θkhk)HΘkηk is still Gaussian, with zero meanand variance ‖(Θkhk)HΘkηk‖2 = ‖(Θkhk)H‖2. Therefore,as shown in (6), the received function can be treated as anapproximate SISO system which can realize single symboldetection.

4) Near Maximum Likelihood Signal Detection: As dis-cussed before (6), νk is assumed to be complex Gaussian,the likelihood function p(sk|yk, ρk) is given by:

p(sk|yk, ρk) =1

π‖Θkhk‖2exp

(−|yk − ρksk|2

‖Θkhk‖2

)(7)

Hence, all the possible modulated symbols related to sk canbe examined by:

sk = argminsk∈A

|yk − ρksk|2 (8)

The value of (8) only depends on sk, which can be determined

TABLE I

LINEAR SIGNAL DETECTION ALGORITHM FOR V-BLAST.

Step 1: Initialization, at k = 11. Channel Power Ordering2. Apply Gaussian approximation in (4);3. Employ the pre-whitening filter in (5) andmatched filter in (6);4. Recover the symbol by (8).

Step 2: Pre-Whitening Detection, at k = 2, 3, · · · , M − 1For k = 2 : M − 1,

1. Remove the past decisions by (3);2. Apply Gaussian approximation in (4);3. Employ the pre-whitening filter in (5) andmatched filter in (6);4. Recover the symbol by (8).

end;Step 3: Tail Detection, at k = M

1. Remove the past decisions by (3);2. Recover the last symbol by ML decoding:sM = argmin

sM∈A|rM − sM hM |2.

by choosing the symbol that can minimize the argument.The overall algorithm is summarized in Tab. I. Note that thecalculation of Λ−1/2

k can be simplified by the matrix inversionlemma [11].

B. Theoretical Performance Analysis

Now, we find the closed-form symbol error rate (SER) ofthe proposed LSD. The SER for the k-th symbol is given by[12] for L-PSK constellations:

SERkL =

1π

∫ (L−1)πL

0

exp

(−Lpskγk

sin2 θ

)dθ (9)

where the received the signal to noise ratio (SNR) for aparticular realization of the fading is give by γk = |ρk|2|s|2

var(νk) =

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‖Θkhk‖2, and Lpsk � sin2 πL . Since the current decision is

made on the basis of past decisions, which become more andmore accurate, we normally have SER1

L≥SER2L≥ · · ·≥SERM

L .The closed-form SER over M transmit symbols can be

approximated calculated as::

SERL =1M

M∑k=1

SERkL (10)

Next, we analyze the diversity order of the proposed LSD.Assuming perfect channel state information at the receiver, andwithout the loss of generality taking BPSK as an example, itcan be upper bounded by:

SERk2 ≤ 1

2exp (−γk) =

12exp

(−‖Θkhk‖2)

(11)

From the Section III-A, we can have the following expression:

‖Θkhk‖2 = hHk Λ−1

k hk

At high SNR, the effect of HHk+1,M Hk+1,M is comparatively

small, and thus HHk+1,M Hk+1,M maybe be ignored in an

asymptotic analysis. Finally, we have the following approx-imation:

‖Θkhk‖2 = σ−2hHk (IM−k − DD+)hk ≈ σ−2 k

M‖hk‖2 (12)

Please refer to Appendix for detailed proof. while hk has sizeN × 1, (11) can be rewritten as:

SERk2 ≤ 1

2exp

(− k

Mσ2

N∑i=1

∣∣αik

∣∣2) (13)

where αik represents the i-th element of h̃k, which is obviously

still i.i.d circularly symmetric complex Gaussian. By averaging(13) over the Rayleigh PDF, we can finally have:

SERk2 � 1

2

(k

Mσ2

)−N

(14)

which shows that the mean SER is approximately inverselyproportional to the signal to noise ratio to N . This demon-strates the importance of the rank since the rank (receivediversity order), M , will determine the slope of the SER curve.As a result, full receive diversity N can be achieved. Also, theanalysis is easily adapted to other constellations.

IV. SIMULATION RESULTS

In all simulation, we consider an uncoded system with fourtransmit antennas and eight receive antennas for V-BLASTover Rayleigh fading channels. Perfect channel knowledgeis assumed at the transmitter and receiver end. The channelemployed here is Rayleigh flat fading. BPSK and QPSKconstellations are applied to achieve rate 4bps/Hz and 8bps/Hz.We use analytical BER for comparison, which can be writtenas:

BERL =1M

M∑k=1

BERkL (15)

where BERkL ≈ 1

LSERkL [5].

The theoretical BER is added for comparison in Fig. 1 and2. We can see that it is almost identical to the simulated

curve. Besides, LSD with power ordering (PO) gives betterperformance than the decoding based on signal arrival (SA).Moreover, as discussed before in Section V, the closed-formBER/SER obtained by PO is better than that by SA, such thatit can be treated as the upper bound of the V-BLAST system.

In Fig. 3 and Fig. 4, it can be observed from the simulatedcurves that there is 1dB loss compared to ML decoder forLSD with either BPSK or QPSK constellations, but it canprovide much better BER than MMSE. when power orderingis applied, it is seen that the LSD with power ordering canoutperform the one that just arranges the sequence of decodingbased on signal arrival about 1dB at BER=10−5. Hence, powerorder is very important for V-BLAST employing LSD. Wecan also see that as there is only 1dB loss compared to MLdecoding, the theoretical BER can be treated indeed as a’close’ upper bound for V-BLAST scheme.

V. CONCLUSIONS

In this paper, we introduce a probabilistic data associationbased signal detector for BLAST schemes over time-selectivefading channels. By employing Gaussian approximation andsignal filtering, a virtual SISO system at the receiver end canbe realized. Regarding the resulting performance, there is onlya small gap for V-BLAST compared to ML. The closed-formSER is given as well for the proposed decoder.

APPENDIX IPROOF OF (12)

As shown in Section III, for convenience, we define D �Hk+1,M and D+ � (HH

k+1,M Hk+1,M )−1HHk+1,M . Note that

since rank(DD+) = rank(HHk+1,M Hk+1,M ) = M − k, and

D+ is a pseudo inverse of matrix D, we have [14]:

Tr(DD+) = rank(DD+) = M − k

It is also known that Tr(DD+) =M−k∑i=1

λi, where λi denotes

the i-th eigenvalue of DD+, and thus the sum of the eigenval-ues in DD+ equal M − k.

Since DD+ is positive-defined, by eigenvalue decomposi-tion, further transformation can be carried out:

IM−k − DD+ = U(IM−k − Π)UH

where U is a unitary matrix and Π �diag{λ1...λM−k, 0...0}. The hermitian of IM−k − DD+ canbe calculated as:

(IM−k − DD+)H(IM−k − DD+) = IM−k − DD+ ≥ 0

Therefore, the eigenvalue of IM−k − DD+ is positive, andfrom

IM−k − Π = diag{1 − λ1, ..., 1 − λM−k, 1...1}we can get:

1 − λi ≥ 0, (i = 1...M − k)

RecallingM−k∑i=1

λi = M − k, we have:

λi = 1, (i = 1...M − k)

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Now, as hk is independent of U, ‖hkU‖2 = ‖hk‖2. We canfinally obtain:

‖Θkhk‖2 = σ−2hHk (IM−k − DD+)hk ≈ σ−2 k

M‖hk‖2

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0 1 2 3 4 5 6 7 8 9 1010

−4

10−3

10−2

10−1

BPSK

SNR [dB]

BE

R

LSD, SAAnalytical LSD, SA LSD, POAnalyticalLSD, PO

Fig. 1. Analytical Performance of the Linear Signal Detector for V-BLAST.

0 2 4 6 8 10 12 1410

−4

10−3

10−2

10−1

100

QPSK

SNR [dB]

BE

R

LSD, SA Analytical LSD, SALSD, POAnalytical LSD, PO

Fig. 2. Analytical Performance of the Linear Signal Detector for V-BLAST.

0 1 2 3 4 5 6 7 8 9 1010

−4

10−3

10−2

10−1

QPSK

SNR [dB]

BE

R

MMSELSD, SALSD, POML

Fig. 3. Simulated Performance of the Linear Signal Detector for V-BLAST.

0 2 4 6 8 10 1210

−4

10−3

10−2

10−1

100

QPSK

SNR [dB]

BE

R

MMSELSD, SALSD, POML

Fig. 4. Simulated Performance of the Linear Signal Detector for V-BLAST.

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