On the Capacity of the Peak Power Constrained Vector Gaussian Channel: An Estimation Theoretic Perspective

TL;DR

This paper investigates the capacity of high-dimensional Gaussian channels with peak power constraints, establishing conditions for optimal single-sphere input distributions and analyzing their scaling behavior as dimension increases.

Contribution

It provides necessary and sufficient conditions for when a single-sphere input distribution is optimal and derives the asymptotic scaling of the maximum radius supporting this optimality.

Findings

The optimal input distribution is supported on a finite number of spheres.

Single-sphere optimality holds below a certain radius  ar{R}_n.

The maximum radius  ar{R}_n scales as  \,\sqrt{n} and its limit is characterized.

Abstract

This paper studies the capacity of an $n$-dimensional vector Gaussian noise channel subject to the constraint that an input must lie in the ball of radius $R$ centered at the origin. It is known that in this setting the optimizing input distribution is supported on a finite number of concentric spheres. However, the number, the positions and the probabilities of the spheres are generally unknown. This paper characterizes necessary and sufficient conditions on the constraint $R$ such that the input distribution supported on a single sphere is optimal. The maximum $\bar{R}_n$, such that using only a single sphere is optimal, is shown to be a solution of an integral equation. Moreover, it is shown that $\bar{R}_n$ scales as $\sqrt{n}$ and the exact limit of $\frac{\bar{R}_n}{\sqrt{n}}$ is found.