MIMO detection employing Markov ChainMonte Carlo

V. Sundaram* and K.P.N.Murthyt* Department Electrical Engineering, University of Notre Dame, IN46556 Indiana, United States of America

t School of Physics, University of Hyderabad, Central University P.O., Hyderabad 500 046, Andhra Pradesh, India

where n is the circularly symmetric channel additive Gaussianwhite noise vector, with independent components each withthe same variance (J'2. For the kth bit of the m th symbol 8 m ,

denoted as bm,k, the MAP detector computes its LY-th valuegiven by

II. PROBLEM FORMULATION

Consider a MIMO system with transmit and receive an­tennas, see Fig. (1). The source bits are encoded and theninterleaved before being mapped to M symbol streams fortransmission. Each symbol stream contains symbols drawnfrom the constellation. For an r x t flat-fading channel H,the received signal y is given by

systems based on sequential Monte Carlo (SMC) method.Dong, Wang and Doucet [12] applied the same method toa BLAST-type receiver and demonstrated that this can signifi­cantly improve the performance of MIMO detectors. Recently,Berouzhny et al [13] have reported improved performanceusing a Gibbs sampler. In this paper, we develop a Soft-In­Soft-Out (SISO) MIMO detector using an MCMC algorithmon a multidimensional lattice.

(2)

(1)y==Hs+n

L -I (P(bm ,k==l jy))mk - og, p(bm,k == Oly)

Fig. 1. MIMO system with transmit and receive antennas

III. MARKOV CHAIN MONTE CARLO METHODS

Markov Chain Monte Carlo (MCMC) methods are a familyof statistical simulation algorithms that help construct anensemble of realizations or states from the desired multivariateprobability distribution. In a typical MCMC method we startfrom an arbitrary state and construct a Makov chain whoseasymptotic part contains states that belong to the desireddistribution. This is ensured by suitably constructing a Markovtransition matrix so that its stationary distribution coincideswith the desired distribution. With easy availability of highperformance computing machines in recent times this methodhas become popular and is being increasingly employed in

I. INTRODUCTION

Multiple-Input-Multiple-Output (MIMO) systems improvethe channel capacity many-fold by the use ofmultiple antennasat transmitter and at receiver[I], [2]. It was shown [1] that thechannel capacity increases linearly with the number of transmitantennas in a rich scattering environment. Because of this ithas become possible to design transmission schemes wherea single data stream is split into several substreams that aresimultaneously transmitted to the available transmit antennas.MIMO forms the core technology for the next generationwireless networks, as seen in the standards IEEE 802.11n(wireless LAN) as well as IEEE 802.16e (wireless MAN).

The optimal MIM0 receiver distinguishes the spatial sig­natures of different transmit substreams as seen at the re­ceiver, while fully exploiting available receive diversity. Thedetection process also accounts for the structure of symbolconstellations. The latter feature implies that a Maximum aPosterior (MAP) receiver is inherently more robust (in termsof error rates) to low-rank channels, as noted in [3]. However,the implementation of an exact MAP detector requires testingof all possible hypotheses in order to compute the reliabilityof each bit, the number of hypotheses being exponential inthe number of transmit antennas and the number of bits.There has therefore been extensive work both in reducing thecomputational complexity of optimal or near-optimal detectors[4], [5], [6] and in devising sub-optimal approaches to MIMOdetection. Some of the latter include a space-time OFE withhard [7] and soft cancellation[8] and the use of iterativereceivers [9], [10]. The major drawback of all the spheredecoding algorithms and their variants is their worst case com­plexity (which is exponential) and the problems encounteredin computing soft values.

Markov Chain Monte Carlo (MCMC) methods essentiallyconsist of drawing samples from a desired probability distribu­tion. Multidimensional systems (such as MIMO) are speciallysuited for MCMC methods, whose complexity is at most poly­nomial with respect to signal dimensions. In a recent paper,Guo and Wang [11] proposed detection methods for MIMO

Abstract-We propose a soft-output detection scheme forMultiple-Input-Multiple-Output (MIMO) systems. The detectoremploys Markov Chain Monte Carlo method to compute bitreliabilities from the signals received and is thus suited for codedMIMO systems. It offers a good trade-off between achievableperformance and algorithmic complexity.

(4)

ex L L p(yISm, 8(-m))p(Sm, 8(-m))STnES11 ) S( -Tn)

(5)

(6)

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•

•

•

•

•Fig. 2. Example of a nearest neighbour jump

•

•

•

•

•

•

•

ex L L p(yISm,8(-m))STnES11 ) S(-Tn)

p(bm,k==lly) == L L p(Sm,8(-m)ly)STnES11 ) S(-Tn)

The final expression in the above is obtained assuming the sumis dominated by those symbol vectors that are most probable.The number of these terms will be denoted by N s • Eq. (5) canbe interpreted as a Monte Carlo integral, see [13]. A similarexpression can be derived for p(bm,k == Oly).

VI. A RANDOM WALK METROPOLIS ALGORITHM ON A

SYMBOL LATTICE

The L-values may thus be computed by a search over thesymbol lattice. In a recent paper [13], a uniform Gibbs sampleris used to perform the search. The algorithm may be improvedby confining the search to those lattice points that are moreprobable. This is captured by the likelihood p(yI8) that canbe sampled using a Metropolis algorithm.

Consider a random walk Metropolis sampler that has thetarget distribution p(yI8). Observing that each vector 8 has atotal of 4t nearest neighbors (see Fig. 2),

and applying Bayes theorem with prior symbol distribution

significant terms

{41 if 8' is a neighbour of 8

Q(8'18) == t

o otherwise

A periodic boundary condition is applied at the lattice edges.At each iteration step, a trial state is uniformly randomly

a variety of fields that include physics, chemistry, biology,economics and engineering see e.g. [14]-[17].

In this paper, we use MCMC method to solve a Bayesianinference problem. To this end, we identify the desired proba­bility distribution (say p(x) of the random variable x) overwhich the inference is to be made. All possible discretevalues that x can take, constitute the state space. We constructa Markov chain of states. The transition probability of theMarkov chain determines the trajectory of the Markov Chainin the state space. Asymptoticlly the Markov chin convergesto the desired disribution. The effectiveness of MCMC metodrelies on the following:

a. the Markov Chain quickly reaches its stationary distribu­tion and

b. the desired inference parameter usually depends only onthe most probable values of the random variable, thatis, those states that are most frequently visited by theMarkov Chain.

Thus, spanning the entire state space that is usually exponentialwith the size ofx is not necessary to extract useful informationabout the distribution.

IV. THE METROPOLIS ALGORITHM

The Metropolis algorithm [18], [19] draws samples froma probability distribution p(x), referred to as the target dis­tribution in this paper. The idea behind this algorithm isto construct a Markov chain whose stationary distributionmatches with the target distribution. Given the current statex a Metropolis sampler draws a candidate state, also calledtrial state Xt. The next state in the Markov chain can be eitherx itself or the trial state Xt. The choice is made randomly bydrawing random numbers. To this end we define a Metropolisacceptance probability given by p == min(l, p(x)/p(Xt)). Ifthe random number drawn is less than p the trial state isaccepted as the next entry in the Markov chain. Otherwise thecurrent state continues and forms the next state. The transitionprobability matrix Q defined by the Metropolis algorithm,satisfies detailed balance condition, given by

p(xi)Q(xjlxi) == p(Xj)Q(xilxj) V i,j (3)

Note that knowledge of the trget distribution is required onlyupto a normalization constant, which makes this algorithmvery useful for simulating equilibrium statistical physics sys­tems where the normalizing partition function is not known.For details on Markov Chain Monte Carlo detection methodssee [20], [21]

V. BIT RELIABILITY ESTIMATION USING MCMC

The posterior probability that bm,k == 1 is given by

p(bm,k == 11y) == L p(Sm,kly)STnEsl1 )

Here SkI) denotes all the symbols from the constellation Sthat have 1 in their kth bit position. Expressing Eq. (4) as themarginalized probability over all other symbols

8(-m) == [So SI ... Sm-l Sm+l ... St-l]

chosen amongst the nearest neighbors of S, and is acceptedas the next state with the Metropolis acceptance probability

. ( p(YIS'))p = nnn 1, p(yIS)

We point out here that since Metropolis algorithm has beendesigned to sample conditional distributions p(yIS, once theMarkov chain runs for a sufficiently long time, the mostfrequently occuring state will be the ML estimate with highprobability. To calculate the L values for each bit, we useequation (4) to constrain the state space for each bit by theset S~1) (ewap. S~O) as shown and pick up other states fromthe MArkov chain described earlier.

The efficiency of a Metropolis sampler depends criticallyon its ability to span important regions of the sample space.This is captured in the acceptance ratio, which is defined asthe number of actual state changes to the total number ofMetropolis attempts. Efficient samplers have acceptance ratiosbetween 0.4 to 0.7.

The argument of the exponential function of the prior distri­bution is sclaled by a constant referred to in statistical physicsliterature as temperature. In several problems, the Metropolisalgorithm can be improved by increasing the temperature. Inthe present case, noise plays the role of temperature in thesystem. Increasing the temperature improves the sample spacecoverage. In our simulations, the sampling temperature wasset to be 10 times that of the ambient noise.

10-4

REFERENCES

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[2] V. Tarokh, N. Seshadri, and A.R. Calderbank, "Space-Time Codes forHigh Data Rate Wireless Communication: Performance Criterion andCode Construction", IEEE Trans. Info. Theory, Vol. 44, No.2, pp.744­765, 1998.

[3] H. Artes, D. Seethaler, F. Hlawatsch, "Efficient detection algorithmsfor MIMO channels: a geometrical approach to approximate MLdetection", IEEE Trans. Sig. Proc., Vol. 51, No. 11, pp.2808-2820,2003.

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[8] Won-Joon Choi, Kok-Wui Cheong and 1. M. Cioffi, "Iterative softinterference cancellation for multiple antenna systems", Proc. IEEEWCNC 2003, Vol. 1, pp.304-309 Sept. 2000.

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[10] Kee-Bong Song and S. A. Mujtaba, "A low complexity space-frequencyBICM MIMO-OFDM system for next-generation WLAN", Proc. IEEEGLOBECOM 2003, Vol. 2, pp.I059-I063 Dec. 2003.

[11] D. Guo and X.Wang, "Blind detection in MIMO systems via sequentialMonte Carlo", IEEE 1. Sel. Areas Commun., Vol. 21, No.2, pp.464473,Apr. 2003.

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VIII. CONCLUSIONS

We have proposed a soft output MIMO detector employing arandom walk Metropolis algorithm. We find that this methodprovides a good tradeoff between computational complexityand soft output reliability. Some interesting possibilities arethe inclusion of a prior in the detection process and reducingthe complexity of each Metropolis step. These are currentlybeing investigated.

1412106 8EblNo (dB)

Me performance with and witholi uniform sampling- --- ,--- - --- -,-- --- ------------,--- ---- -----

10-0 '---_---'-__---'--__L---_-----'-- .L--_---'

o

10° -

10 '

10-2

0:::Wal

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--+-- Me with 250 runs, non-uniform, 10*No-lI!- Me with 250 runs, uniform sampling

-- soft-ML

Fig. 3. Simulation of RWMA algorithm on a 3x3 16-QAM system

VII. SIMULATION RESULTS

We now present some simulation results on a 3 x 3 MIMOsystem, i.e., with 3 transmit and 3 receive antennas. A packetof 64 bytes was encoded using a rate-1/2 convolutional codewith the generator matrix G == [133, 171] (in octal notation).After interleaving, the bits are mapped to a (gray-coded)16-QAM constellation. The MIMO channel is assumed tobe i.i.d. Rayleigh fading, and changes with every channeluse. At the receiver, the perfect CSI is assumed and theMetropolis sampler estimates the bit reliabilities (LLRs) usingthe Random Walk Metropolis Algorithm (RWMA) describedabove. The total transmit is set to unity with equal powerallocation across streams. The key parameters that determinesthe detector complexity is the number of Metropolis iterations(denoted by R), and the number of significant terms used(denoted by N s ). In all the simulations, we have set N s == 1,that corresponds to the max-log-MAP approximation. We havealso compared the results with uniform sampling. In Figure 3we have plotted the bit error rate versus the per antenna SNR.As the results in Figure 3 show, the frame error rate with theproposed algorithm is superior to that of uniform sampling,and seems to improve when R is increased.

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E. Teller, "Equation of state calculation by fast computing machines",1. Chern. Phys. Vol. 21, pp. 1087, 1953.

[19] G. Bhanot, "The Metropolis Algorithm", Rep. Prog. Phys. Vol. 51,pp.429, 1988.

[20] S.A. Laraway and B. Farhang-Boroujeny, "Implementation ofa MarkovChain Monte Carlo Based Multiuser/MIMO Detector," Submitted toIEEE Trans. On Signal Processing, Oct. 2006.

[21] H. Zhu, B. Farhang-Boroujeny, and R-R. Chen, "On performance ofsphere decoding and Markov chain Monte Carlo detection methods,"IEEE Signal Processing Letters, Oct. 2005, pp. 669-672.