The Shannon's mutual information of a multiple antenna time and frequency dependent channel: an ergodic operator approach

TL;DR

This paper analyzes the mutual information of a multi-antenna frequency-selective channel using ergodic operator theory, showing its convergence to a deterministic limit as the number of antennas grows large.

Contribution

It introduces an ergodic operator framework to evaluate the mutual information in large multi-antenna channels, linking it to random matrix theory results.

Findings

Mutual information converges to a deterministic limit with increasing antennas.

The ergodic operator approach aligns with random matrix theory predictions.

Analysis of the Stieltjes transform elucidates the asymptotic behavior.

Abstract

Consider a random non-centered multiple antenna radio transmission channel. Assume that the deterministic part of the channel is itself frequency selective, and that the random multipath part is represented by an ergodic stationary vector process. In the Hilbert space $l^2({\mathbb Z})$, one can associate to this channel a random ergodic self-adjoint operator having a so-called Integrated Density of States (IDS). Shannon's mutual information per receive antenna of this channel coincides then with the integral of a $\log$ function with respect to the IDS. In this paper, it is shown that when the numbers of antennas at the transmitter and at the receiver tend to infinity at the same rate, the mutual information per receive antenna tends to a quantity that can be identified and, in fact, is closely related to that obtained within the random matrix approach. This result can be obtained by analyzing the behavior of the Stieltjes transform of the IDS in the regime of the large numbers of antennas.