Capacity of Finite-State Channels with Time-Invariant Deterministic Feedback

We consider channel coding with feedback for the general case where the feedback may be an arbitrary deterministic function of the output samples. Under the assumption that the channel states take values in a finite alphabet, we find an achievable rate and an upper bound on the capacity. We conclude by showing that when the channel is indecomposable, and has no intersymbol interference, its capacity is given by the limit of the maximum of the (normalized) directed information between the input X^N and the output Y^N, i.e. C = lim_Nrarrinfin/1N max I(X^N rarr Y^N), where the maximization is over the causal conditioning probability Q(x^N||k^N-) defined in this paper