On entropy-constrained Gaussian channel capacity via the moment problem

TL;DR

This paper investigates the capacity of Gaussian channels with input entropy constraints, revealing that only up to three moments can be matched by low-entropy discrete distributions, with implications for low SNR regimes.

Contribution

It characterizes the entropy-constrained Gaussian channel capacity at low SNR and introduces a novel moment matching limitation for discrete distributions with small entropy.

Findings

Capacity characterization at low SNR with entropy constraints

Maximum of three moments matched by low-entropy discrete distributions

Implications for channel capacity and distribution design

Abstract

We study the capacity of the power-constrained additive Gaussian channel with an entropy constraint at the input. In particular, we characterize this capacity in the low signal-to-noise ratio regime at small entropy. This follows as a corollary of the following general result on a moment matching problem: We show that for any continuous random variable with finite moments, the largest number of initial moments that can be matched by a discrete random variable of sufficiently small but positive entropy is three.