Channel Coding for Gaussian Channels with Mean and Variance Constraints

TL;DR

This paper characterizes the optimal performance of Gaussian channels with mean and variance constraints, introducing a novel achievability scheme using mixtures of uniform distributions on spheres and providing bounds on output distribution ratios.

Contribution

It presents a new achievability scheme for Gaussian channels with mean and variance constraints using mixtures of sphere-based codewords and proves bounds on output distribution ratios.

Findings

Achieves matching converse and achievability bounds for the problem.

Introduces a novel mixture distribution scheme for codewords.

Provides high-probability bounds on output distribution ratios.

Abstract

We consider channel coding for Gaussian channels with the recently introduced mean and variance cost constraints. Through matching converse and achievability bounds, we characterize the optimal first- and second-order performance. The main technical contribution of this paper is an achievability scheme which uses random codewords drawn from a mixture of three uniform distributions on $(n-1)$-spheres of radii $R_1, R_2$ and $R_3$, where $R_i = O(\sqrt{n})$ and $|R_i - R_j| = O(1)$. To analyze such a mixture distribution, we prove a lemma giving a uniform $O(\log n)$ bound, which holds with high probability, on the log ratio of the output distributions $Q_i^{cc}$ and $Q_j^{cc}$, where $Q_i^{cc}$ is induced by a random channel input uniformly distributed on an $(n-1)$-sphere of radius $R_i$. To facilitate the application of the usual central limit theorem, we also give a uniform $O(\log n)$ bound, which holds with high probability, on the log ratio of the output distributions $Q_i^{cc}$ and $Q^*_i$, where $Q_i^*$ is induced by a random channel input with i.i.d. components.