Maximum Likelihood Sequence Estimation (MLSE) is widely used in digital communication systems to estimate the transmitted data sequence observed over noisy ISI channels. For an unknown channel, many algorithms have been developed to estimate the sequence and/or identify the channel blindly. In this paper we focus on direct sequence estimation for unknown, fast time-varying channels.
Per-Survivor Processing (PSP) [1] has been proposed for MLSE with fast time-varying channels. In [2], we describe an approach to MLSE based on probabilistic modeling of the fast time-varying channel. We formulate the MAP sequence estimator, assign a prior distribution to the parameters of a time-varying channel model, and marginalize over the channel parameters. For temporally independent Gaussian channels, an optimum Viterbi algorithm is derived. For Gauss-Markov channels, a PSP approach based on the generalized Viterbi algorithm (GVA) [3] is described which retains a fixed number of survivors (L) per trellis state. We show that the proposed method approaches that of ML exhaustive search for reasonably small values of L.
In [4], we employ the EM algorithmic approach to solve the MAP problem described in [2]. This approach is similar to other EM based approaches to direct sequence estimation (e.g. [5,6,7,8]), but is unique in its focus on exploiting prior information on fast time-varying channel models to improve performance. The EM method for iterative solution of Maximum Likelihood (ML) and MAP estimation problems for continuous parameters was introduced into the signal processing community by Dempster, et al. [9]. With EM, convergence and initialization are important considerations. Wu [10] proved convergence to a local minimum or saddle point of the negative log likelihood function for the continuous parameter case. This holds true for the MAP problem posterior density [11]. For the problem of discrete parameter (e.g. sequence) estimation, convergence is less understood. In [11] it is claimed that for certain discrete parameter estimation EM problems, including the one addressed in [4], EM is guaranteed to converge to a stationary point of the posterior density. In this paper we show that this claim is not true, and that EM is not guaranteed to converge to a stationary point. Nonetheless, numerous researchers have shown that EM can be used effectively for sequence estimation.
In this paper we describe and contrast direct and EM iterative algorithms for MAP sequence estimation with unknown, fast time-varying channels.