Capacity and power scaling laws for finite antenna amplify-and-forward relay networks

A novel framework is presented that can be used to study the capacity and power scaling of linear multiple-input multiple-output (MIMO) d×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 framework can be applied to systems with any per-hop channel fading distribution provided the expected logarithm of the channel matrices' norms are finite; in this contribution all of our results relate specifically to 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, 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 present some numerical examples to highlight the theory.