Sparse Array Beampattern Synthesis via Alternating Direction Method of Multipliers

Abstract

In this paper, we devise beampattern synthesis algorithms for sparse arrays using the alternating direction method of multipliers (ADMM). Unlike the usual weighted ${l} 1$ norm, we utilize the ${l} p$ norm of the array weight vector, where $0 , as the objective function to enhance its sparsity for arbitrary array configurations. To solve the resultant nonconvex and nonlinear optimization problem, we introduce auxiliary variables to decouple the array weight vector in the objective function from the complicated constraints on main lobes and sidelobes, and then, the array weight vector and auxiliary variables are updated alternately via ADMM. To determine the array weight vector with the ${l} p$ norm, we analyze the convexity or concavity of the subfunction related to each weight element using its derivatives. On the other hand, we divide the objective function of auxiliary variables into multiple nonlinear subfunctions, each of which is only dependent of the magnitude of the corresponding auxiliary variable and is calculated in parallel via analyzing simplified two-sided constraints. Furthermore, we extend our methodology to the symmetric excitation case with symmetric array configurations. Numerical examples show that the proposed methods can obtain satisfactory radiation pattern with fewer antennas than the existing techniques and are applicable for arbitrary or symmetric array configurations.