Sparse Array Beampattern Synthesis via Two-Stage Penalty Dual Decomposition Method

In this work, a new two-stage penalty dual decomposition (TSPDD) method is devised for the power mask-constrained sparse array beampattern synthesis problem. Unlike the usual approximation of
$l_{0}$
-quasi-norm with weighted
$l_{1}$
-norm or
$l_{\mathrm {p}}$
-quasi-norm (
$0 < p < 1$
), we directly utilize the objective function of
$l_{0}$
-quasi-norm to solve this problem without any approximation, so it can obtain a sparser solution than some other methods. The TSPDD solve the augmented Lagrangian problem via dual decomposition (DD)-type methods. It is an iteration algorithm. In the first stage, we update the penalty parameters, while in the second stage, we update the dual variables. Furthermore, in order to further enhance the sparsity of the solution, we extended the algorithm TSPDD to extended two-stage penalty dual decomposition (E-TSPDD) by discarding the array elements with smaller weights and replacing two close array elements with one. Finally, we considered the application of the proposed method in the sparse array synthesis problem with mutual coupling. Numerical examples show that the proposed methods have a good convergence and performances. It can achieve satisfactory radiation pattern with fewer antennas than the existing techniques and can handle the mutual coupling effect in the sparse array synthesis task.