A random search framework for convergence analysis of distributed beamforming with feedback

The focus of this work is on the analysis of transmit beamforming schemes with a low-rate feedback link in wireless sensor/relay networks, where nodes in the network need to implement beamforming in a distributed manner. Specifically, the problem of distributed phase alignment is considered, where neither the transmitters nor the receiver has perfect channel state information, but there is a low-rate feedback link from the receiver to the transmitters. In this setting, a framework is proposed for systematically analyzing the performance of distributed beamforming schemes. To illustrate the advantage of this framework, a simple adaptive distributed beamforming scheme that was recently proposed by Mudambai is studied. Two important properties of the received signal magnitude function are derived. Using these properties and the systematic framework, it is shown that the adaptive distributed beamforming scheme converges both in probability and in mean. Furthermore, it is established that the time required for the adaptive scheme to converge in mean scales linearly with respect to the number of sensor/relay nodes.