Fast and Simple Gradient Projection Algorithms for Phase-Only Beamforming

Phase-only beamforming (POB) plays an important role in modern radar and communication systems. The NP-hard nature of POB makes it difficult to be solved for large-scale antenna arrays. This paper studies two fast and simple GP algorithms for the problem of POB. First we give a new interpretation of the GP method under the framework of sequential quadratic programming (SQP). Then we prove that the GP operation gives a strict global minimizer of the subproblem of SQP. Moreover, the larger the step size in GP operation, the smaller the minimum value of the subproblem. Based on this observation we design two variable step size GP (VSGP) algorithms that improve the convergence speed of the GP method significantly. Furthermore, the VSGP algorithms have a simple structure, making them easy to be implemented in chips. Finally, we derive a necessary and sufficient condition for the Karush-Kuhn-Tucker (KKT) points of the POB problem. We also prove that, for any initialization, all limit points of the iterates generated by the VSGP algorithms converge to the KKT points. Simulation results show that the proposed algorithms outperform the state-of-the-art algorithms in terms of convergence speed and computational cost.