The Capacity and Optimal Signaling for Gaussian MIMO Channels Under Interference Constraints (full version)

TL;DR

This paper derives the optimal transmit covariance for Gaussian MIMO channels under power and interference constraints, providing closed-form solutions, iterative algorithms, and insights into regime transitions and matrix properties.

Contribution

It introduces a comprehensive solution framework for optimal signaling in MIMO channels with interference constraints, including closed-form, iterative, and rank bounds, advancing understanding of optimal transmission strategies.

Findings

Closed-form solutions for optimal covariance matrices.

An iterative bisection algorithm with proven convergence.

Insights into power and interference regime transitions.

Abstract

Gaussian MIMO channel under total transmit and interference power constraints (TPC and IPC) is considered. A closed-form solution for the optimal transmit covariance matrix in the general case is obtained using the KKT-based approach (up to dual variables). While closed-from solutions for optimal dual variables are possible in special cases, an iterative bisection algorithm (IBA) is proposed to find the optimal dual variables in the general case and its convergence is proved for some special cases. Numerical experiments illustrate its efficient performance. Bounds for the optimal dual variables are given, which facilitate numerical solutions. An interplay between the TPC and IPC is studied, including the transition from power-limited to interference-limited regimes as the total transmit power increases. Sufficient and necessary conditions for each constraint to be redundant are given. A number of explicit closed-form solutions are obtained, including full-rank and rank-1 (beamforming) cases as well as the case of identical eigenvectors (typical for massive MIMO settings). A bound on the rank of optimal covariance is established. A number of unusual properties of optimal covariance matrix are pointed out.