On The Capacity of Gaussian MIMO Channels Under Interference Constraints (full version)

TL;DR

This paper derives closed-form solutions for the optimal transmit covariance in Gaussian MIMO channels with interference constraints, revealing new insights into capacity limits, optimal precoding, and spectrum sharing conditions.

Contribution

It provides novel closed-form solutions for MIMO capacity under interference constraints, including special cases and the role of whitening filters, advancing understanding beyond water-filling strategies.

Findings

Closed-form solutions for optimal covariance matrices are derived.

Capacity can be zero despite non-zero transmit power under certain conditions.

Spectrum sharing is characterized by a simple rank condition.

Abstract

Gaussian MIMO channel under total transmit and multiple interference power constraints (TPC and IPCs) is considered. A closed-form solution for its optimal transmit covariance matrix is obtained in the general case (up to dual variables). A number of more explicit closed-form solutions are obtained in some special cases, including full-rank and rank-1 (beamforming) solutions, which differ significantly from the well-known water-filling solutions (e.g. signaling on the channel eigenmodes is not optimal anymore and the capacity can be zero for non-zero transmit power). A whitening filter is shown to be an important part of optimal precoding under interference constraints. Capacity scaling with transmit power is studied: its qualitative behaviour is determined by a natural linear-algebraic structure induced by MIMO channels of multiple users. A simple rank condition is given to characterize the cases where spectrum sharing is possible. An interplay between the TPC and IPCs is investigated, including the transition from power-limited to interference-limited regimes. A number of unusual properties of an optimal covariance matrix under IPCs are pointed out and a bound on its rank is established. Partial null forming known in the adaptive antenna array literature is shown to be optimal from the information-theoretic perspective as well in some cases.