Convergence analyses and comparisons of Markov chain Monte Carlo algorithms in digital communications

Recently, Markov chain Monte Carlo (MCMC) methods have been applied to the design of blind Bayesian receivers in a number of digital communications applications. The salient features of these MCMC receivers include the following: a) They are optimal in the sense of achieving minimum symbol error rate; b) they do not require the knowledge of the channel states, nor do they explicitly estimate the channel by employing training signals or decision-feedback; and c) they are well suited for iterative (turbo) processing in coded systems. In this paper, we investigate the convergence behaviors of several MCMC algorithms (both existing and new ones) in digital communication applications. The geometric convergence property of these algorithms is established by considering only the chains or the marginal chains corresponding to the transmitted digital symbols, which take values from a finite discrete set. We then focus on three specific applications, namely, the MCMC decoders in AWGN channels, ISI channels, and CDMA channels. The convergence rates for these algorithms are computed for small simulated datasets. Different convergence behaviors are observed. It is seen that differential encoding, parameter constraining, collapsing, and grouping are efficient ways of accelerating the convergence of the MCMC algorithms, especially in the presence of channel phase ambiguity.