High-SNR Second-Order Statistics of Amplify-and-Forward Relaying with Variable Gains and Multiple Hops

We analyze the high-SNR behavior of the level crossing rate and average fade duration for a variable-gain amplify-and-forward relaying system composed of an arbitrary number of hops under Rayleigh fading. Most studies on the second-order statistics of multi-hop amplify-and-forward relaying assume fixed gains at the relays and analyze the end-to-end channel gain (from source to destination). Such analysis leads to the so-called cascaded product channel, being mathematically attractive, yet ignoring the accumulation of noise along the hops. Here, in turn, we take the noise into account by analyzing the end-to-end SNR, which ultimately governs the system performance. Also, aiming at the benchmark for transparent relaying, we assume variable gains at the relays. We obtain simple closed-form asymptotic expressions for the investigated second-order statistics and, as a byproduct, for the associated outage probability.