On the Finiteness of the Capacity of Continuous Channels

TL;DR

This paper establishes mild conditions under which the capacity of continuous, possibly non-linear, memoryless channels with additive noise is finite and achievable, broadening understanding of channel capacity limits.

Contribution

It introduces a novel sufficient condition ensuring the convergence of differential entropies, leading to new insights on when channel capacity is finite for general noise channels.

Findings

Capacity is finite under broad conditions, including infinite second moments.

Finiteness holds for most setups with non-linear, memoryless channels.

Differential entropy convergence is key to establishing capacity finiteness.

Abstract

Evaluating the channel capacity is one of many key problems in information theory. In this work we derive rather-mild sufficient conditions under which the capacity is finite and achievable. These conditions are derived for generic, memoryless and possibly non-linear additive noise channels. The results are based on a novel sufficient condition that guarantees the convergence of differential entropies under point-wise convergence of Probability Density Functions. Perhaps surprisingly, the finiteness of channel capacity holds for the majority of setups, including those where inputs and outputs have possibly infinite second-moments.