A Scalable Feedback Mechanism for Distributed Nullforming With Phase-Only Adaptation

This paper considers a problem of distributed nullforming, in which multiple wireless transmitters steer a null toward a designated receiver by only adjusting their carrier phases. Since each transmitter transmits at full power, the system maximizes “power pooling” gains for cooperative communication or jamming, while simultaneously protecting a designated receiver. Analysis in a noiseless setting shows that, while the received power at the designated receiver, as a function of the transmitted phases, is nonconvex with multiple critical points, all of its local minima are also global minima. This implies that a null can be formed using a distributed, scalable protocol based on gradient descent: each transmitter adapts its phase based only on aggregate feedback broadcast by the receiver (so that feedback overhead does not increase with the number of transmitters), along with an estimate of its own channel gain (which can be obtained, e.g., via reciprocity). Simulations show that the convergence rate actually improves with the number of transmitters, and that the algorithm is robust to noise, substantial channel estimation errors, and oscillator drift.