Distributed nullforming without prior frequency synchronization

We present a novel approach to the problem of distributed nullforming where a set of transmitters cooperatively transmit a common message signal in such a way that their individual transmissions precisely cancel each other at a designated receiver. Under our approach, each transmitter iteratively makes an adjustment to the phase and frequency of its transmitted RF signal, by effectively implementing an algorithm to reduce the amplitude of the overall received signal to zero. We show that this algorithm can be implemented in a purely distributed fashion at each transmitter as each transmitter needs only an estimate of its own channel gain to the receiver, and a feedback signal from the receiver, that is common across all the transmitters. This is an important advantage of our approach and assures its scalability; in contrast any non-iterative approach to the nullforming problem requires that each transmitter know every other transmitter's channel gain. We prove analytically that the algorithm practically, globally converges to a null at the designated receiver. By practical convergence we mean that (i) the algorithm always converges to a stationary trajectory, (ii) that though some of these trajectories may not correspond to a null, those that do not are locally unstable, while those that do are locally stable. We also present numerical simulations to illustrate the robustness of this approach. A key novelty of this paper is that unlike its predecessors it does not assume prior frequency synchronization among the transmitters.