Single-Letter Bounds on the Feedback Capacity of Unifilar Finite-State Channels

Upper and lower bounds on the feedback capacity of unifilar finite-state channels (FSCs) are derived. The upper bound is derived using a new technique, called the Q-contexts, which is based on a construction of a directed graph that is used to quantize recursively the receiver's output sequences to a finite set of contexts. For any choice of Q-graph, the feedback capacity is bounded by a single-letter expression, Cfb <= sup I(X, S; Y vertical bar Q), where the supremum is over PX vertical bar S,Q. and the distribution of (S,Q) is their stationary distribution. The bound is tight for all unifilar FSCs where feedback capacity is known: channels where the state is a function of the outputs, the trapdoor channel, Ising channels, the no-consecutive-ones input-constrained erasure channel and for the memoryless channel. The upper bound indicates that a single-letter expression might exist for the capacity of finite-state channels with or without feedback which are based on a construction of auxiliary random variable with memory, such as Q-graph, and not with i.i.d distribution. The lower bound provides a sufficient condition for the optimality of the upper bound, however, it is formulated such that independent lower bounds on feedback capacity may be calculated. The efficiency of these bounds is demonstrated by deriving a new capacity result for the dicode erasure channel (DEC). The upper bound also serves as a non-trivial bound on the capacity of channels without feedback, a problem that is still open.