The MIMO Iterative Waterfilling Algorithm

TL;DR

This paper provides a comprehensive analysis of the MIMO game for arbitrary channel matrices, establishing conditions for the uniqueness of Nash equilibria and convergence of iterative waterfilling algorithms, including new technical results for singular matrices.

Contribution

It introduces new theoretical tools and conditions that extend the analysis of MIMO waterfilling algorithms to singular and rectangular channel matrices, and proposes a modified game with milder convergence conditions.

Findings

Uniqueness and convergence conditions are more restrictive for tall, possibly singular matrices.

A new expression for the MIMO waterfilling projection valid for singular matrices.

A modified game and algorithm with milder conditions and similar equilibrium performance.

Abstract

This paper considers the non-cooperative maximization of mutual information in the vector Gaussian interference channel in a fully distributed fashion via game theory. This problem has been widely studied in a number of works during the past decade for frequency-selective channels, and recently for the more general MIMO case, for which the state-of-the art results are valid only for nonsingular square channel matrices. Surprisingly, these results do not hold true when the channel matrices are rectangular and/or rank deficient matrices. The goal of this paper is to provide a complete characterization of the MIMO game for arbitrary channel matrices, in terms of conditions guaranteeing both the uniqueness of the Nash equilibrium and the convergence of asynchronous distributed iterative waterfilling algorithms. Our analysis hinges on new technical intermediate results, such as a new expression for the MIMO waterfilling projection valid (also) for singular matrices, a mean-value theorem for complex matrix-valued functions, and a general contraction theorem for the multiuser MIMO watefilling mapping valid for arbitrary channel matrices. The quite surprising result is that uniqueness/convergence conditions in the case of tall (possibly singular) channel matrices are more restrictive than those required in the case of (full rank) fat channel matrices. We also propose a modified game and algorithm with milder conditions for the uniqueness of the equilibrium and convergence, and virtually the same performance (in terms of Nash equilibria) of the original game.