Beamforming via Nonconvex Linear Regression

Impulsive processes frequently occur in many fields, such as radar, sonar, communications, audio and speech processing, and biomedical engineering. In this paper, we propose a nonconvex linear regression (NLR) based minimum dispersion beamforming technique for impulsive signals to achieve significant performance improvement over the conventional minimum variance beamformer. The proposed beamformer minimizes the $\ell_{p}$-norm of the output with $p<1$ subject to a linear distortionless response constraint, resulting in a difficult nonconvex and nonsmooth optimization problem. The constrained optimization problem is first reduced to a multivariate linear regression via constraint elimination. As a major contribution of this paper, a coordinate descent algorithm (CDA) is devised for solving the resultant NLR problem of $\ell_{p}$-minimization with $p<1$ at a computational complexity of ${\cal O}(MN^{2})$, where $M$ is the number of sensors and $N$ is the sample size. At each inner iteration of the CDA, an efficient algorithm is designed to find the global minimum of each subproblem of univariate linear regression. The convergence of the CDA is analyzed. The NLR beamformer with a single constraint is further generalized to the case of multiple linear constraints, which is robust against model mismatch. Simulation results demonstrate the superior performance of nonconvex optimization based beamformer.