A Single-Letter Upper Bound on the Feedback Capacity of Unifilar Finite-State Channels

Abstract

An upper bound on the feedback capacity of unifilar finite-state channels (FSCs) is derived. A new technique, called the $Q$ -context mapping, is based on a construction of a directed graph that is used for a sequential quantization of 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, $C_{\mathrm {fb}}\leq \sup I(X,S;Y|Q)$ , where the supremum is over $p(x|s,q)$ and the distribution of $(S,Q)$ is their stationary distribution. It is shown that 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 the memoryless channel. Its efficiency is also demonstrated by deriving a new capacity result for the dicode erasure channel; the upper bound is obtained directly from the above-mentioned expression and its tightness is concluded with a general sufficient condition on the optimality of the upper bound. This sufficient condition is based on a fixed point principle of the BCJR equation and, indeed, formulated as a simple lower bound on feedback capacity of unifilar FSCs for arbitrary $Q$ -graphs. This upper bound indicates that a single-letter expression might exist for the capacity of finite-state channels with or without feedback based on a construction of auxiliary random variable with specified structure, such as the $Q$ -graph, and not with i.i.d distribution. The upper bound also serves as a non-trivial bound on the capacity of channels without feedback, a problem that is still open.