Adaptive Distributed Beamforming for Amplify-and-Forward Relay Networks: Convergence Analysis

For wireless amplify-and-forward (AF) relay networks, this work focuses on the convergence analysis of adaptive distributed beamforming schemes that can be reformulated as local random search algorithms via a random search framework. It is proved that under two sufficient conditions: 1) the objective function of the random search algorithm is continuous and all its local maxima are global maxima in the considered feasible set, and 2) the origin is an interior point within the support of the probability measure for the random perturbation, the corresponding adaptive distributed beamforming schemes converge almost surely. While the second sufficient condition can be controlled by system designer and satisfied with relative ease, the first sufficient condition initially seems strict. Surprisingly, further analysis on the signal-to-noise ratio (SNR) functions in AF relay networks with individual and total power constraints demonstrates that, in both scenarios, local maxima are global maxima and hence, the first sufficient condition is satisfied. Finally, the proposed framework was extended to analyze adaptive distributed beamforming schemes in an asynchronous setting, and simulation results were provided to further validate our analysis.