Fast Unit-Modulus Least Squares With Applications in Beamforming

Abstract

Unit-modulus least squares (ULS) problems arise in many applications, including phase-only beamforming, sensor network localization, synchronization, phase retrieval, and radar code design. ULS formulations can always be recast as unit-modulus quadratic programs, to which semidefinite relaxation (SDR) can be applied, and is often the state-of-the-art approach. However, SDR lifts the problem dimension (i.e., the number of variables) from $N$ to $N^2$ , which drastically increases the memory burden and computational cost when the problem size is already large—e.g., when designing phase-only beamformer weights for massive multiple-input-multiple-output systems. This paper focuses on scalable first-order algorithms for the ULS problem and some of its variants. It advocates using simple gradient projection (GP) as a starting point for solving the ULS problem, establishes global convergence of GP to a Karush–Kuhn–Tucker point for this NP-hard problem, and bounds its iteration complexity. Then it proposes ULS extensions tailored to reflect practical beamformer design objectives, bringing in and exploiting new degrees of freedom to improve the beampattern designs. Simple variants of GP are proposed to deal with these extended ULS problems. Simulations are used to showcase the effectiveness of the proposed algorithms in both the plain ULS problem and in the context of phase-only beamforming.