Capacity of Markov channels with partial state feedback

We study the capacity of Markov channels with causal deterministic partial (quantized) state feedback. We assume the feedback channel to be memoryless, the channel state process to be Markovian, belong to a finite set, and the state and observation transitions to satisfy some general mixing conditions. For such channels, we obtain a single-letter characterization for the capacity with feedback. We further show that for every epsilon > 0, there exists a finite length memory (sliding) encoder structure that leads to an e-optimal capacity; hence practically optimal performance can be achieved. We show that the non-linear filter generating the conditional state density provides the sufficient statistic for the optimal coding scheme.