Capacity of Compound MIMO Gaussian Channels with Additive Uncertainty

TL;DR

This paper investigates the capacity of compound MIMO Gaussian channels with bounded spectral norm uncertainty, revealing a hidden convexity in the optimal transmit design and proving a conjecture about the channel capacity duality.

Contribution

It demonstrates that the optimal transmit covariance matrix diagonalizes the nominal channel and establishes zero duality gap, enabling convex optimization approaches for this problem.

Findings

Optimal transmit design diagonalizes the nominal channel.

The duality gap between compound and min-max capacity is zero.

The problem can be transformed into a convex optimization problem.

Abstract

This paper considers reliable communications over a multiple-input multiple-output (MIMO) Gaussian channel, where the channel matrix is within a bounded channel uncertainty region around a nominal channel matrix, i.e., an instance of the compound MIMO Gaussian channel. We study the optimal transmit covariance matrix design to achieve the capacity of compound MIMO Gaussian channels, where the channel uncertainty region is characterized by the spectral norm. This design problem is a challenging non-convex optimization problem. However, in this paper, we reveal that this problem has a hidden convexity property, which can be exploited to map the problem into a convex optimization problem. We first prove that the optimal transmit design is to diagonalize the nominal channel, and then show that the duality gap between the capacity of the compound MIMO Gaussian channel and the min-max channel capacity is zero, which proves the conjecture of Loyka and Charalambous (IEEE Trans. Inf. Theory, vol. 58, no. 4, pp. 2048-2063, 2012). The key tools for showing these results are a new matrix determinant inequality and some unitarily invariant properties.