On the Distortion of the Eigenvalue Spectrum in MIMO Amplify-and-Forward Multi-Hop Channels

TL;DR

This paper analyzes how the eigenvalue spectrum distortion affects the capacity of multi-hop MIMO channels, revealing conditions under which capacity scales linearly with the number of antennas and degrees of freedom are preserved.

Contribution

It provides an asymptotic capacity analysis for multi-hop MIMO channels with relay clusters, establishing scaling laws and spectral behavior under large system limits.

Findings

Capacity scales linearly with min{n_s, n_d} under certain scaling conditions.

Almost all spatial degrees of freedom vanish with sublinear scaling.

Point-to-point MIMO capacity is approached with slightly faster than linear scaling.

Abstract

Consider a wireless MIMO multi-hop channel with n_s non-cooperating source antennas and n_d fully cooperating destination antennas, as well as L clusters containing k non-cooperating relay antennas each. The source signal traverses all L clusters of relay antennas, before it reaches the destination. When relay antennas within the same cluster scale their received signals by the same constant before the retransmission, the equivalent channel matrix H relating the input signals at the source antennas to the output signals at the destination antennas is proportional to the product of channel matrices H_l, l=1,...,L+1, corresponding to the individual hops. We perform an asymptotic capacity analysis for this channel as follows: In a first instance we take the limits n_s->infty, n_d->infty and k->infty, but keep both n_s/n_d and k/n_d fixed. Then, we take the limits L->infty and k/n_d->infty. Requiring that the H_l's satisfy the conditions needed for the Marcenko-Pastur law, we prove that the capacity scales linearly in min{n_s,n_d}, as long as the ratio k/n_d scales at least linearly in L. Moreover, we show that up to a noise penalty and a pre-log factor the capacity of a point-to-point MIMO channel is approached, when this scaling is slightly faster than linear. Conversely, almost all spatial degrees of freedom vanish for less than linear scaling.