Rate distortion functions for finite-state finite-alphabet Markov sources

A lower bound to the rate-distortion function of finite-alphabet sources with memory is derived for the class of balanced distortion measures. For finite-state finite-alphabet Markov sources, sufficient conditions are given for the existence of a strictly positive average distortion such that equals its lower bound for . The bound is evaluated for the Hamming and Lee distortion measures and is identical to the corresponding bound for memoryless sources having the same entropy and alphabet. These results are applied to yield a simple proof of the converse of the noisy-channel coding theorem for sources satisfying the sufficient conditions for equality with the lower bound and channels with memory. is evaluated explicitly for the special case of the binary asymmetric Markov source.