Coding Theorem and Converse for Abstract Channels with Time Structure and Memory

TL;DR

This paper establishes a coding theorem and converse for a broad class of stationary channels with time structure and memory, relaxing previous conditions and including important models like Gaussian channels.

Contribution

It introduces weaker conditions on channel output memory and relaxes measurability requirements, broadening the applicability of coding theorems to more realistic channel models.

Findings

Proves coding theorem under total ergodicity with weaker memory conditions.

Shows $\\psi$-mixing is equivalent to finite output memory for Gaussian channels.

Provides a corrected proof of a key lemma in Kadota and Wyner's work.

Abstract

A coding theorem and converse are proved for a large class of abstract stationary channels with time structure including the result by Kadota and Wyner (1972) on continuous-time real-valued channels as special cases. As main contribution the coding theorem is proved for a significantly weaker condition on the channel output memory - called total ergodicity w.r.t. finite alphabet block-memoryless input sources - and under a crucial relaxation of the measurability requirement for the channel. These improvements are achieved by introducing a suitable characterization of information rate capacity. It is shown that the $\psi$-mixing output memory condition used by Kadota and Wyner is quite restrictive and excludes important channel models, in particular for the class of Gaussian channels. In fact, it is proved that for Gaussian (e.g., fading or additive noise) channels the $\psi$-mixing condition is equivalent to finite output memory. Further, it is demonstrated that the measurability requirement of Kadota and Wyner is not satisfied for relevant continuous-time channel models such as linear filters, whereas the condition used in this paper is satisfied for these models. Moreover, a weak converse is derived for all stationary channels with time structure. Intersymbol interference as well as input constraints are taken into account in a general and flexible way, including amplitude and average power constraints as special case. Formulated in rigorous mathematical terms complete, explicit, and transparent proofs are presented. As a side product a gap in the proof of Kadota and Wyner - illustrated by a counterexample - is closed by providing a corrected proof of a lemma on the monotonicity of some sequence of normalized mutual information quantities. An abstract framework is established to treat discrete- and continuous-time channels with memory and arbitrary alphabets in a unified way.