Hidden Markov models (HMMs) are a powerful tool for modeling stochastic random processes. They are general enough to model with high accuracy a large variety of processes and are relatively simple allowing us to compute analytically many important parameters of the process which are very difficult to calculate for other models (such as complex Gaussian processes). Another advantage of using HMMs is the existence of powerful algorithms for fitting them to experimental data and approximating other processes. In this paper, we demonstrate that communication channel fading can be accurately modeled by HMMs, and we find closed-form solutions for the probability distribution of fade duration and the number of level crossings.

Conventional differential detection is known to perform poorly on the fast-fading Rayleigh-fading channel due to the rapid variation of the channel state. The technique of multiple symbol differential detection (MSDD) improves the error performance by detecting over a sequence of symbols and exploiting the inherent correlation of the fading process. To further reduce the error rate, we need to introduce some form of coding. Successful coding for the Rayleigh-fading channel requires interleaving to decorrelate the channel-a conflicting requirement to MSDD. We propose a solution consisting of interleaving blocks of L M-ary phase-shift keying (MPSK) symbols that are multiple symbol differentially detected. The blocks are treated as elements of an M/sup L/ symbol alphabet over which a multilevel code based on geometrically uniform partitions is defined.

The majority of previous analytical studies of signal-space coding techniques (includes trellis and block codes) on the Rayleigh-fading channel have assumed ideal interleaving. The effect of finite interleaving on the performance of different coding schemes has been studied only by simulation In this paper we first derive a maximum likelihood (ML) decoder for codewords transmitted over a noninterleaved Rayleigh flat fading channel, followed by an exact expression for the pairwise error event probability of such a decoder. It includes phase shift keying (PSK), quadrature amplitude modulation (QAM) signal sets, trellis coded modulation (TCM) and block coded modulation (BCM) schemes, as well as coherent (ideal channel state information) and partially coherent (e.g., differential, pilot tone, etc.) detection. We derive an exact expression for the pairwise event probability in the case of very slow fading-i.e., the fading experienced by all the symbols of the codeword is highly correlated. We also show that the interleaving depth required to optimize code performance for a particular minimum fading bandwidth can be approximated by the first zero of the fading channel's auto-correlation function.