Returning to Shannon's Original Meaning

TL;DR

This work argues that ergodicity is essential to the Shannon capacity framework and critiques the Verdú–Han general formula when applied to non-ergodic channels. It shows that, for ergodic memoryless channels, the classic capacity $\max_{P_X} I(X;Y)$ is recovered, while non-ergodic (stateful) channels require state-conditioned capacity expressions such as $\min_i \max_{P_X} I(X;Y|S=s_i)$. The paper further contends that slow fading channels should be analyzed via outage capacity $C_\epsilon$ rather than a strict Shannon sense capacity, and emphasizes restoring Shannon’s original meaning by aligning theory with practical ergodicity concepts. Overall, it argues for a careful separation between ergodic Shannon-sense capacity and engineering notions like outage capacity, and it provides corrections to the interpretation of Verdú–Han’s general formula in non-ergodic settings.

Abstract

Shannon theory is revisited to show that ergodicity is an indispensable element of channel capacity. The generalized channel capacity $C=\sup_{\bm{X}}\underline{I}(\bm{X}; \bm{Y})$ is checked with a negative conclusion and the popular assertion "the capacity of a slow fading channel is zero in strict Shannon sense" is found to be conceptually wrong.