Optimal detection of discrete Markov sources over discrete memoryless channels-applications to combined source-channel coding

The authors consider the problem of detecting a discrete Markov source which is transmitted across a discrete memoryless channel. Two maximum a posteriori (MAP) formulations are considered: (i) a sequence MAP detection in which the objective is to determine the most probable transmitted sequence given the observed sequence and (ii) an instantaneous MAP detection which is to determine the most probable transmitted symbol at time n given all the observations prior to and including time n. The solution to the first problem results in a “Viterbi-like” implementation of the MAP detector (with Large delay) while the latter problem results in a recursive implementation (with no delay). For the special case of the binary symmetric Markov source and binary symmetric channel, simulation results are presented and an analysis of these two systems yields explicit critical channel bit error rates above which the MAP detectors become useful. Applications of the MAP detection problem in a combined source-channel coding system are considered. Here, it is assumed that the source is highly correlated and that the source encoder (a vector quantizer (VQ)) fails to remove all of the source redundancy. The remaining redundancy at the output of the source encoder is referred to as the “residual” redundancy. It is shown, through simulation, that the residual redundancy can be used by the MAP detectors to combat channel errors. For small block sizes, the proposed system beats Farvardin and Vaishampayan´s channel-optimized VQ by wide margins. Finally, it is shown that the instantaneous MAP detector can be combined with the VQ decoder to form an approximate minimum mean-squared error decoder