Markov sources achieve the feedback capacity of finite-state machine channels

The feedback capacity of a finite-state machine channel is achieved by a feedback-dependent Markov source with the same memory length as the channel. The optimal feedback is captured by the conditional probabilities of the channel states given all previous channel outputs, i.e., by the forward coefficients in the Bahl, Cocke, Jelinek and Raviv (1974) algorithm. We formulate the optimization of the feedback-dependent Markov source distribution as an average-reward-per-stage stochastic control problem, and solve it numerically using dynamic programming algorithms.