Distributed adaptive beamforming in wireless sensor networks is the attempt to achieve phase coherency in a transmission among a set of distributed transmitters to a remote receiver. With this technique it is possible to decrease the transmission power required by each individual node, to increase the transmission range or to improve the robustness of transmission. Traditional algorithmic solutions require significant computational capabilities of nodes and utilise, for instance, CDMA techniques together with Phase locked loops to each individual transmitter. These requirements typically surcharge the computational capabilities of simple sensor nodes. Recently, however, a computationally cheap randomised scheme for phase synchronisation was presented in [1].

We derive an asymptotically sharp bound on the synchronisation speed of this randomised black box optimisation technique for closed-loop feedback based distributed adaptive beamforming in wireless sensor networks. We also show that the feedback function that guides this synchronisation process is weak multimodal.

Given this knowledge that no local optimum exists, we consider an approach to locally compute the phase offset of each individual carrier signal. With this design objective an asymptotically optimal algorithm is derived. Additionally, we discuss the concept to reduce the optimisation time and energy consumption by hierarchically clustering the network into subsets of nodes that achieve beamforming successively over all clusters. For the approaches discussed we demonstrate their practical feasibility in simulations and experiments.

[1] R. Mudumbai, G. Barriac, and U. Madhow, "On the feasibility of distributed beamforming in wireless networks," IEEE Trans. on Wireless Commun., vol. 6, no. 5, pp. 1754-1763.