A RANDOMIZATION METHOD FOR QUASI MAXIMUM LIKELIHOOD DECODING Amin Mobasher * , Mahmoud Taherzadeh * , Renata Sotirov † , and Amir K. Khandani * * Electrical and Computer Engineering Department † Department of Mathematics & Statistics University of Waterloo, Waterloo, ON, Canada The University of Melbourne, Parkville, VIC, Australia Email:{amin, taherzad, khandani}@cst.uwaterloo.ca Email:rsotirov@ms.unimelb.edu.au Abstract— In Multiple-Input Multiple-Output (MIMO) systems, Maximum-Likelihood (ML) decoding is equivalent to £nding the closest lattice point in an N dimensional complex space. In [1], we have proposed several quasi- maximum likelihood relaxation models for decoding in MIMO systems based on semi-de£nite programming. In this paper, we propose randomization algorithms that £nd a near-optimum solution of the decoding problem by exploring the solution of the corresponding semi-de£nite relaxations. 1 INTRODUCTION Recently, there has been a considerable interest in Multi- Input Multi-Output (MIMO) antenna systems due to achieving a very high capacity compared to single- antenna systems [2]. In MIMO systems, a vector is trans- mitted by transmit antennas. In the receiver, a corrupted version of this vector affected by the channel noise and fading is received. Decoding concerns the operation of receiving the transmitted vector from the received signal. This problem is usually expressed in terms of ”lattice decoding” which is known to be NP-hard. Quasi-maximum likelihood detection is a near opti- mum algorithm for lattice decoding based on a binary programming formulation and semi-de£nite relaxation [1], [3]. More precisely, the distance minimization in the Euclidean space is formulated in terms of a bi- nary quadratic minimization problem. Then, the resulting problem is transformed into a relaxation problem us- ing Semi-De£nite Programming (SDP). The solution for the distance minimization problem is a rank-one binary matrix. However, the rank-one constraint is removed in the relaxation problem. Therefore, the solution for the relaxation problems is not necessarily a binary rank-one matrix. This solution is changed to a 0–1 rank-one matrix through a randomization procedure. The extreme points of the feasible set for the relaxation problem are the binary rank-one matrices. The randomization procedure determinants some of the extreme points by using a solution of the SDP relaxation. Among these extreme points, the one which results in the smallest value for the distance-minimization objective function is chosen as the solution point. The randomization procedure in [3] is based on {−1, 1} elements. Usually, communication applications deal with 0–1 vectors, and the formulation of the problem with {−1, 1} elements is not always a simple task. Here, we propose a method that depends on the bit values, {0, 1}. Also, with a smaller number of iterations it achieves a better performance compared to those which are relying on {−1, 1} elements. 2 MIMO SYSTEM MODEL A MIMO system with ˜ N transmit antenna and ˜ M receive antenna is modelled as ˜ y =  SNR ˜ M ˜ E sav ˜ H˜ x +˜ n, (1) where ˜ H =  ˜ h ij  is the ˜ M × ˜ N channel matrix with independent, identically distributed complex Gaussian random variables with zero mean and unit variance, ˜ n is an ˜ M × 1 complex additive white Gaussian noise vector with zero mean and unit variance elements, and ˜ x is an ˜ N ×1 vector whose components are the signals sent from each transmit antenna and selected from a complex set ˜ S = {˜ s 1 , ˜ s 2 , ··· , ˜ s K } with the average energy ˜ E sav . The parameter SNR in (1) is the Signal to Noise Ratio (SNR) per receive antenna. Noting ˜ x i ∈S , for i =1, ··· , ˜ N we have ˜ x i = u i (1)˜ s 1 + u i (2)˜ s 2 + ··· + u i (K)˜ s K , (2) where u i (j ) ∈{0, 1} and K  j=1 u i (j )=1 ∀ i =1, ··· , ˜ N. (3) Let u =  u 1 (1) ··· u 1 (K) ··· u N (1) ··· u N (K)  T and N = ˜ N . Using the equations in (2) and (3), the transmitted vector is expressed as ˜ x = ˜ Su, (4) where ˜ S = I N ⊗ [˜ s 1 , ··· , ˜ s K ] is an N × NK matrix of coef£cients, I N is an N × N Identity matrix, ⊗ is the tensor product, and u is an NK × 1 binary vector PROCEEDINGS OF THE 9TH CANADIAN WORKSHOP ON INFORMATION THEORY, MONTREAL, QUEBEC, JUNE 5 - 8, 2005 135