Upper Bounds on the Capacities of Noncontrollable Finite-State Channels with/without Feedback

TL;DR

This paper develops computable upper bounds on the capacities of noncontrollable finite-state channels by transforming the problem into a stochastic control framework and solving it with value iteration.

Contribution

It introduces a novel approach that incorporates delayed channel states into the input, enabling the derivation of upper bounds via ARSCP and finite-state approximations.

Findings

Upper bounds are achievable by conditional Markov sources.

The bounds can be computed through a convergent value iteration algorithm.

The method applies to channels with and without feedback.

Abstract

Noncontrollable finite-state channels (FSCs) are FSCs in which the channel inputs have no influence on the channel states, i.e., the channel states evolve freely. Since single-letter formulae for the channel capacities are rarely available for general noncontrollable FSCs, computable bounds are usually utilized to numerically bound the capacities. In this paper, we take the delayed channel state as part of the channel input and then define the {\em directed information rate} from the new channel input (including the source and the delayed channel state) sequence to the channel output sequence. With this technique, we derive a series of upper bounds on the capacities of noncontrollable FSCs with/without feedback. These upper bounds can be achieved by conditional Markov sources and computed by solving an average reward per stage stochastic control problem (ARSCP) with a compact state space and a compact action space. By showing that the ARSCP has a uniformly continuous reward function, we transform the original ARSCP into a finite-state and finite-action ARSCP that can be solved by a value iteration method. Under a mild assumption, the value iteration algorithm is convergent and delivers a near-optimal stationary policy and a numerical upper bound.