Capacity Scaling in MIMO Systems with General Unitarily Invariant Random Matrices

TL;DR

This paper analyzes how the capacity of MIMO systems scales with the number of antennas, revealing that mutual information loss depends on system dimensions and the distribution of channel matrices, especially under Haar distribution assumptions.

Contribution

It provides a novel analysis of capacity scaling in MIMO systems with unitarily invariant random matrices, including explicit formulas for mutual information loss and asymptotic behavior.

Findings

Mutual information loss depends only on system dimensions and the left-singular vectors matrix.

Ergodic rate loss for Haar-distributed matrices is given by a double sum formula.

Normalized rate loss converges to the binary entropy function as system dimensions grow large.

Abstract

We investigate the capacity scaling of MIMO systems with the system dimensions. To that end, we quantify how the mutual information varies when the number of antennas (at either the receiver or transmitter side) is altered. For a system comprising $R$ receive and $T$ transmit antennas with $R>T$, we find the following: By removing as many receive antennas as needed to obtain a square system (provided the channel matrices before and after the removal have full rank) the maximum resulting loss of mutual information over all signal-to-noise ratios (SNRs) depends only on $R$, $T$ and the matrix of left-singular vectors of the initial channel matrix, but not on its singular values. In particular, if the latter matrix is Haar distributed the ergodic rate loss is given by $\sum_{t=1}^{T}\sum_{r=T+1}^{R}\frac{1}{r-t}$ nats. Under the same assumption, if $T,R\to \infty$ with the ratio $\phi\triangleq T/R$ fixed, the rate loss normalized by $R$ converges almost surely to $H(\phi)$ bits with $H(\cdot)$ denoting the binary entropy function. We also quantify and study how the mutual information as a function of the system dimensions deviates from the traditionally assumed linear growth in the minimum of the system dimensions at high SNR.