Probabilistic Analysis of the Semidefinite Relaxation Detector in Digital Communications

We consider the problem of detecting a vector of symbols that is being transmitted over a fading multiple–input multiple–output (MIMO) channel, where each symbol is an M–th root of unity for some fixed M ≥ 2. Although the symbol vector that minimizes the error probability can be found by the so–called maximum–likelihood (ML) detector, its computation is intractable in general. In this paper we analyze a popular polynomial–time heuristic, called the semidefinite relaxation (SDR) detector, for the problem and establish its first non–asymptotic performance guarantee. Specifically, in the low signal–to–noise ratio (SNR) region, we show that for any M ≥ 2, the SDR detector will yield a constant factor approximation to the optimal log–likelihood value with a probability that increases exponentially fast to 1 as the channel size increases. In the high SNR region, it is known that for M = 2, the SDR detector will yield an exact solution to the ML detection problem with a probability that converges to 1. We refine this result by establishing the rate of convergence. Our work can be viewed as an average– case analysis of a certain SDP relaxation, and the input distribution we use is motivated by physical considerations. Our results also refine and extend those in previous work, which are all asymptotic in nature and apply only to the problem of detecting binary (i.e., when M = 2) vectors. In particular, our results can give better insight into the performance of the SDR detector in practical settings.