The structure of encoders for channels with feedback

We extend our previous work on channels with feedback to cases with partial feedback. Specifically [2], [3] treated the case of channels with memory and full output feedback. In this paper we treat the case of general channels with memory and partial deterministic feedback. One of the main difficulties in this problem has to do with the fact that the transmitter and the receiver may have different information about the state of the channel. We show how to choose appropriate sufficient statistics at both the transmitter and receiver. To this end we introduce the notion of joint sufficiency. These jointly sufficient statistics give insight into the structure of the channel encoder. We then formulate the capacity problem as an infinite horizon average cost stochastic control problem. The resulting Bellman equation leads to an implicit single-letter characterization of the capacity.