MIMO detection employing Markov Chain Monte Carlo
aa r X i v : . [ c s . G L ] M a y MIMO detection employing Markov Chain Monte Carlo
V. Sundaram ⋆ and K.P.N.Murthy † ⋆ Department Electrical Engineering,University of Notre Dame,IN46556 Indiana, United States of Americaand † School of Physics, University of Hyderabad,Central University P.O.,Gachibowli, Hyderabad 500 046,Andhra Pradesh, India (Dated: October 29, 2018)We propose a soft-output detection scheme for Multiple-Input-Multiple-Output(MIMO) systems. The detector employs Markov Chain Monte Carlo method tocompute bit reliabilities from the signals received and is thus suited for coded MIMOsystems. It offers a good trade-off between achievable performance and algorithmiccomplexity.
I. INTRODUCTION
Multiple-Input-Multiple-Output (MIMO) systems improve the channel capacity many-fold by the use of multiple antennas at transmitter and at receiver[1, 2]. It was shown[1] that the channel capacity increases linearly with the number of transmit antennas in arich scattering environment. Because of this it has become possible to design transmissionschemes where a single data stream is split into several substreams that are simultaneouslytransmitted to the available transmit antennas. MIMO forms the core technology for thenext generation wireless networks, as seen in the standards IEEE 802.11n (wireless LAN)as well as IEEE 802.16e (wireless MAN).The optimal MIMO receiver distinguishes the spatial signatures of different transmitsubstreams as seen at the receiver, while fully exploiting available receive diversity. Thedetection process also accounts for the structure of symbol constellations. The latter featureimplies that a Maximum ` a Posterior (MAP) receiver is inherently more robust (in terms oferror rates) to low-rank channels, as noted in [3]. However, the implementation of an exactMAP detector requires testing of all possible hypotheses in order to compute the reliabilityof each bit, the number of hypotheses being exponential in the number of transmit antennasand 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 MIMO detection. Some of the latter include a space-time DFE withhard [7] and soft cancellation[8] and the use of iterative receivers [9, 10]. The major drawbackof all the sphere decoding algorithms and their variants is their worst case complexity (whichis exponential) and the problems encountered in computing soft values.Markov Chain Monte Carlo (MCMC) methods essentially consist of drawing samplesfrom a desired probability distribution. Multidimensional systems (such as MIMO) arespecially suited for MCMC methods, whose complexity is at most polynomial with respectto signal dimensions. In a recent paper, Guo and Wang [11] proposed detection methods forMIMO systems based on sequential Monte Carlo (SMC) method. Dong, Wang and Doucet[12] applied the same method to a BLAST-type receiver and demonstrated that this cansignificantly improve the performance of MIMO detectors. Recently, Berouzhny et al [13]have reported improved performance using a Gibbs sampler. In this paper, we develop aSoft-In-Soft-Out (SISO) MIMO detector using an MCMC algorithm on a multidimensionallattice.
II. PROBLEM FORMULATION
Consider a MIMO system with transmit and receive antennas, see Fig. (1). The sourcebits are encoded and then interleaved before being mapped to M symbol streams for trans-mission. Each symbol stream contains symbols drawn from the constellation . For a flat-fading channel H , the received signal y is given by y = H s + n (1)where n is the circularly symmetric channel additive Gaussian white noise vector, withindependent components each with the same variance σ . For the k th bit of the m th symbol hpFIG. 1: MIMO system with transmit and receive antennas s m , denoted as b m,k , the MAP detector computes its L th -value given by L m,k = log p ( b m,k = 1 | y ) p ( b m,k = 0 | y ) ! (2) III. MARKOV CHAIN MONTE CARLO METHODS
Markov Chain Monte Carlo (MCMC) methods are a family of statistical simulation algo-rithms that help construct an ensemle of realizations or states from the desired multivariateprobability distribution. In a typical MCMC method we start from an arbitrry state andconstruct a Makov chain whose asymptotic part contains states that belong to the desireddistribution. This is ensured by suitably constructing a Markov transition matrix so thatits stationary distribution coincides with the desired distribution. With easy availability ofhigh performance computing machines in recent times this method has become popular andis being increasingly employed in a variety of fields that include physics, chemistry, biology,economics and engineering see e.g. [14].In this paper, we use MCMC method to solve a Bayesian inference problem. To this end,we identify the desired probability distribution (say ρ ( x ) of the random variable x ) over whichthe inference is to be made. All possible discrete values that x can take, constitute the statespace. We construct a Markov chain of states. The transition probability of the Markovchain determines the trajectory of the Markov Chain in the state space. Asymptoticlly theMarkov chin converges to the desired disribution. The effectiveness of MCMC metod relieson the following:a. the Markov Chain quickly reaches its stationary distribution andb. the desired inference parameter usually depends only on the most probable valuesof the random variable, that is, those states that are most frequently visited by theMarkov Chain.Thus, spanning the entire state space that is usually exponential with the size of x is notnecessary to extract useful information about the distribution. IV. THE METROPOLIS ALGORITHM
The Metropolis algorithm [15] draws samples from a probability distribution ρ ( x ), referredto as the target distribution in this paper. The idea behind this algorithm is to constructa Markov chain whose stationary distribution matches with the target distribution. Giventhe current state x a Metropolis sampler draws a candidate state, also called trial state x t .The next state in the Markov chain can be either x itself or the trial state x t . The choice ismade randomly by drawing random numbers. To this end we define a Metropolis acceptanceprobabiity given by p = min(1 , ρ ( x ) /ρ ( x t )). If the random number drawn is less than p thetrial state is accepted as the next entry in the Markov chain. Otherwise the current statecontinues and forms the next state. The transition probability matrix Q defined by theMetropolis algorithm, satisfies detailed balance condition, given by ρ ( x i ) Q ( x j | x i ) = ρ ( x j ) Q ( x i | x j ) ∀ i, j (3)Note that the knowledge of the trget distribution is required only upto a normalization con-stant, which makes this algorithm very useful for simulating equilibrium statistical physicssystems where the normalizing partition function is not known. V. BIT RELIABILITY ESTIMATION USING MCMC
The posterior probability that b m,k = 1 is givn by p ( b m,k = 1 | y ) = X S m ∈ S (1) k p ( S m,k | y ) (4)Here S (1) k denotes all the symbols from the constellation S that have 1 in their k th bitposition. Expressing 4 as the marginalized probability over all other symbols S ( − m ) = [ S S · · · S m − S m +1 · · · ]and applying Bayes theorem with prior symbol distribution p ( b m,k = 1 | y ) = X S m ∈ S (1) k X S ( − m ) p ( S m , S ( − m ) | y ) ∝ X S m ∈ S (1) k X S ( − m ) p ( y | S m , S ( − m ) | y ) p ( S m , S ( − m ) ) ∝ X S m ∈ S (1) k X S ( − m ) p ( y | S m , S ( − m ) ) ≈ X sinificant terms p ( y | S m , S ( − m ) ) (5)The final expression in the above is obtained assuming the sum is dominated by those symbolvectors that are most probable. The number of these terms will be denoted by N s . Eq. (5)can be interpreted as a Monte Carlo integral, see [13]. A similar expression can be derivedfor p ( b m,k = 0 | y ). VI. A RANDOM WALK METROPOLIS ALGORITHM ON A SYMBOLLATTICE
The L-values may thus be computed by a search over the symbol lattice. In a recentpaper [13], a uniform Gibbs sampler is used to perform the search. The algorithm may beimproved by confining the search to those lattice points that are more probable. This iscaptured by the likelihood p ( y | S ) that can be sampled using a Metropolis algorithm.Consider a random walk Metropolis sampler that has the target distribution p ( y | S ).Observing that each vector S has a total of 4 M neighbors (see Fig. 2), FIG. 2: Example of a nearest neighbour jump Q ( S ′ | S ) = M if S ′ is a neighbour of S S , and is accepted as the nextstate with the Metropolis acceptance probability p = min , p ( y | S ′ ) p ( y | S ) ! VII. SIMULATION RESULTS
We now present some simulation results on a 3 × i.e. , with 3 transmitand 3 receive antennas. A packet of 64 bytes was encoded using a rate-1 / G = [133 , R ),and the number of significant terms used (denoted by N s ). In all the simulations, we haveset N s = 1, that corresponds to the max-log-MAP approximation. We have also comparedthe results with uniform sampling. As the results in Figure 3 show, the frame error ratewith the proposed algorithm is superior to that of uniform sampling, and seems to improvewhen R is increased.The efficiency of a Metropolis sampler depends critically on its ability to efficiently spanthe sample space. This is captured in the acceptance ratio, which is defined as the number ofnumber of actual state changes to the total number of Metropolis attempts. Efficient sam-plers have acceptance ratios between 0 . .
7. The argument of the exponential functionof the prior distribution is sclaled by a constant referred to in statistical physics literature astemperature. In several problems, the Metropolis algorithm can be improved by increasingthe temperature. In the present case, noise plays the role of temperature in the system.
FIG. 3: Simulation of RWMA algorithm on a 3x3 16-QAM system
Increasing the temperature improves the sample space coverage. In our simulations, thesampling temperature was set to be 10 times that of the ambient noise.
VIII. CONCLUSIONS
We have proposed a soft output MIMO detector employing a random walk Metropolisalgorithm. We find that this method provides a good tradeoff between computational com-plexity and soft output reliability. Some interesting possibilities are the inclusion of a priorin the detection process and reducing the complexity of each Metropolis step. These arecurrently being investigated. [1] G.J. Foschini,
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