Convergence of a Class of Decentralized Beamforming Algorithms

One of the key issues in decentralized beamforming is the need to phase-align the carriers of all the sensors in a wireless sensor network. Recent work in this area has shown the viability of certain methods that incorporate single-bit feedback from a beacon. This paper analyzes the behavior of the method (showing conditions for convergence in distribution and also giving a concrete way to calculate the final distribution of the convergent ball) and then generalizes the method in three ways. First, by incorporating both negative and positive feedback it is possible to double the convergence rate of the algorithm without adversely effecting the final variance. Second, a way of reducing the amount of energy required (by reducing the number of transmissions needed for convergence) is shown; its convergence and final variance can also be conveniently described. Finally, a wideband analog is proposed that operates in a decentralized manner to align the time delay (rather than the phase) between sensors.