Capacity and Power Scaling Laws for Finite Antenna MIMO Amplify-and-Forward Relay Networks

TL;DR

This paper introduces a new framework using random dynamical systems and Lyapunov exponents to analyze capacity and power scaling in finite antenna MIMO amplify-and-forward relay networks, applicable to various fading conditions.

Contribution

It develops a novel analytical approach modeling relay networks as RDS and links Lyapunov exponents to power and capacity trajectories across the network.

Findings

Power at each node follows a deterministic path governed by the maximum Lyapunov exponent.

Eigenchannel capacities follow deterministic trajectories determined by their Lyapunov exponents.

Lyapunov exponents relate to capacity divergence and power costs for multiplexed streams.

Abstract

In this paper, we present a novel framework that can be used to study the capacity and power scaling properties of linear multiple-input multiple-output (MIMO) $d\times d$ antenna amplify-and-forward (AF) relay networks. In particular, we model these networks as random dynamical systems (RDS) and calculate their $d$ Lyapunov exponents. Our analysis can be applied to systems with any per-hop channel fading distribution, although in this contribution we focus on Rayleigh fading. Our main results are twofold: 1) the total transmit power at the $n$th node will follow a deterministic trajectory through the network governed by the network's maximum Lyapunov exponent, 2) the capacity of the $i$th eigenchannel at the $n$th node will follow a deterministic trajectory through the network governed by the network's $i$th Lyapunov exponent. Before concluding, we concentrate on some applications of our results. In particular, we show how the Lyapunov exponents are intimately related to the rate at which the eigenchannel capacities diverge from each other, and how this relates to the amplification strategy and number of antennas at each relay. We also use them to determine the extra cost in power associated with each extra multiplexed data stream.