On the Symmetries and the Capacity Achieving Input Covariance Matrices of Multiantenna Channels

TL;DR

This paper investigates how the symmetries of multiantenna channels determine the structure of capacity-achieving input covariance matrices, providing unified proofs and new insights into optimal input strategies based on symmetry groups.

Contribution

It introduces a unified approach using Haar measure to identify the structure of optimal covariance matrices, simplifying proofs and extending classical capacity theorems.

Findings

Isotropic input is optimal for channels with multiple symmetries.

The structure of capacity-achieving matrices depends primarily on channel symmetries.

Provides a unified framework for classical and new capacity theorems.

Abstract

In this paper we study the capacity achieving input covariance matrices of a single user multiantenna channel based solely on the group of symmetries of its matrix of propagation coefficients. Our main result, which unifies and improves the techniques used in a variety of classical capacity theorems, uses the Haar (uniform) measure on the group of symmetries to establish the existence of a capacity achieving input covariance matrix in a very particular subset of the covariance matrices. This result allows us to provide simple proofs for old and new capacity theorems. Among other results, we show that for channels with two or more standard symmetries, the isotropic input is optimal. Overall, this paper provides a precise explanation of why the capacity achieving input covariance matrices of a channel depend more on the symmetries of the matrix of propagation coefficients than any other distributional assumption.