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

A lower bound to the rate-distortion function
R(D)
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
D_c
such that
R(D)
equals its lower bound for
0 \leqq D \leqq D_c
. 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.
D_c
is evaluated explicitly for the special case of the binary asymmetric Markov source.