Maximum likelihood detection of MIMO system using second order ...

6. CONCLUSIONSIn this paper, we proposed a new ML detection method for theMIMO channel. This method is based on relaxing the MAX-CUT problem and solving the resulting system by SOCP. We haveshown that the MAX-CUT (equivalently ML detection) problemcan be formulated into SOCP problem. The advantage of the proposedmethod over the SDP relaxation is that the difference in thenumber of variables between SDP relaxation and SOCPas£increasesbecomes large. Hence MAX-CUT SOCP is numerically moreefficient. Although, in general, the lower bound computed bySOCP relaxation is not as good as a bound obtained with SDPrelaxation in theory, they are often close to each other [6]. Theproposed method has three significant advantages over the spheredecoding technique.1) The proposed method is polynomial time irrespective of thevalue of the SNR as opposed to sphere decoding which has exponentialcomplexity for low SNR values.2) The SOCP has no heuristic parameters to set (unlike the spheredecoder in which the optimal radius has to be selected in a heuristicway).3) The sphere decoder complexity,¢¡¤£ ¥ , is at each time instant,while there is no such problem in SOCP approach.Moreover our approach does not require number of the receive antennasto be greater than number of the transmit antennas as theBLAST receiver does. Simulations results in figure 2 shows thatSOCP gives very close approximation to the exact ML and thatthere is almost no difference between the performance of the SDPand SOCP method. Simulations are performed for two transmitand two receive antennas. In general for large£the saving of CPUtime gets larger for SOCP in comparison to SDP method.[2] G.J. Foschini, ”Layered space-time architecture for wirelesscommunication in a fading environment when using multielementantenna ”, Bell labs Tech. Journal, vol.1, no.2, pp.41-59, 1996.[3] H. Vikalo and B. Hassibi, ”Maximum likelihood sequence detectionfor multiple antenna system over dispersive channelsvia sphere decoding ”, Eurasip Journal on App. Signal Processing, no.5, pp.525-531, May, 2002.[4] M.X. Goemans and D.P. Williamson, ”Improved approximationalgorithms for maximum cut and satisfiability problemsusing semidefinite programming ” Int. Journal of ACM, no.42,pp.1115-1145, 1995.[5] B. Mohar and S. Poljak, ”Eigenvalues and max-cut problem”Czechoslavak mathematical journal, no.40, pp.343-352, 1990.[6] S. Kim and M. Kojima, ”Second order cone programming relaxationof non-convex quadratic optimization problems” Optimizationmethods and software no.15, pp.201-224, 2001.[7] M.S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, ”Applicationsof second order cone programming, Tech. report January31, 1998.[8] C. Helmberg and F. Rendl , ”Solving quadratic (0,1) problemsby semidefinite programs and cutting planes”, Mathematicalprogramming, vol.82, pp. 219-315, 1998.[9] M. Junger and G. Rinaldi, ”Relaxation of Max-Cut problemand computations of spin glass ground states”, Tech. reportno. 97.300, 1997.[10] Y. E. Nesterov and A. S. Nemirovskii, ”Interior point polynomialalgorithms in convex programming”, SIAM studies inApplied Mathematics, SIAM, Philadelphia, PA, 1994. .10 0BER versus SNRexact MLSDP MLSOCP ML10 −1BER10 −210 −310 −40 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5SNR in dBFig. 2. Av. BER for N=2, M=2 vs ¢¡¤£¦¥¨§©¦ .7. REFERENCES[1] I.E. Talatar, ”Capacity of multi-antenna Gaussian channels ”,European Trans. on Telecommunications, vol.6,no., pp.585-595, no.11, 1999.