Distortion Limited Amplify-and-forward Relay Networks and the $\epsilon$-critical Phase Transition

We study amplify-and-forward (AF) relay networks operating with source and relay amplifier distortion, where the distortion dominates the noise power. The diversity order is shown to be $0$ for fixed-gain (FG) and $1$ for variable-gain (VG) if distortion occurs at the relay; if distortion occurs only at the source, the diversity order will be $1$ for both. With $\epsilon_\beta=N_0/\eta_\beta$ ($N_0$ the noise power, $\eta_\beta$ the distortion power at node $\beta\in\{S,R\}$, the source or relay), we demonstrate the emergence of what we call an $\epsilon$-critical signal-to-noise plus distortion ratio (SNDR) threshold (a threshold that emerges when $\min\{\epsilon_\beta\}$ becomes small) for both forwarding protocols. We show that crossing this threshold in distortion limited regions will cause a phase transition (a dramatic drop) in the network's outage probability. Thus, small reductions in the required end-to-end transmission rate can have significant reductions in the network's outage probability.