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

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 d×d antenna amplify-and-forward relay networks. In particular, we model these networks as random dynamical systems and calculate their d Lyapunov exponents. Our analysis can be applied to systems with any perhop channel fading distribution; although in this contribution, we focus on Rayleigh fading. Our main results are twofold: 1) the total transmit power at the nth node will follow a deterministic trajectory through the network governed by the network's maximum Lyapunov exponent and 2) the capacity of the ith eigenchannel at the nth node will follow a deterministic trajectory through the network governed by the network's ith 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 the number of antennas at each relay. We also use them to determine the extra cost in power associated with each extra multiplexed data stream.