On the Capacity of Vector Gaussian Channels With Bounded Inputs

TL;DR

This paper characterizes the capacity of MIMO Gaussian channels with bounded inputs, deriving conditions for optimal input distributions and providing closed-form solutions under certain constraints, with implications for practical signaling strategies.

Contribution

It generalizes existing methods to higher dimensions, identifies the structure of capacity-achieving inputs, and offers bounds for non-identity channel matrices.

Findings

Optimal input distribution support is finite sets of hyper-spheres.

Constant amplitude signaling achieves capacity when antennas are large and average power is relaxed.

Finite discrete support approximates Gaussian input capacity in memoryless channels.

Abstract

The capacity of a deterministic multiple-input multiple-output (MIMO) channel under the peak and average power constraints is investigated. For the identity channel matrix, the approach of Shamai et al. is generalized to the higher dimension settings to derive the necessary and sufficient conditions for the optimal input probability density function. This approach prevents the usage of the identity theorem of the holomorphic functions of several complex variables which seems to fail in the multi-dimensional scenarios. It is proved that the support of the capacity-achieving distribution is a finite set of hyper-spheres with mutual independent phases and amplitude in the spherical domain. Subsequently, it is shown that when the average power constraint is relaxed, if the number of antennas is large enough, the capacity has a closed form solution and constant amplitude signaling at the peak power achieves it. Moreover, it will be observed that in a discrete-time memoryless Gaussian channel, the average power constrained capacity, which results from a Gaussian input distribution, can be closely obtained by an input where the support of its magnitude is a discrete finite set. Finally, we investigate some upper and lower bounds for the capacity of the non-identity channel matrix and evaluate their performance as a function of the condition number of the channel.