Quadratically Constrained Minimum Dispersion Beamforming via Gradient Projection

Abstract

A quadratically constrained minimum dispersion (QCMD) beamformer that is robust against model uncertainties is devised for non-Gaussian signals. Different from the minimum variance based beamformers, the QCMD beamformer minimizes the $\ell_{p}$ -norm $(p\geq 1)$ of the output while constraining the magnitude response of any steering vector within a spherical uncertainty set to exceed unity. A gradient projection algorithmic framework is proposed to efficiently solve the resulting convex optimization problem instead of directly applying the standard optimization algorithm which has a high computational complexity. In each iteration, the gradient projection updates the solution along the gradient direction and projects it back to the constraint set. Importantly, a closed-form expression of the projection onto the constraint set is derived, which only needs a low complexity of ${\cal O}(M)$ with $M$ being the sensor number. Therefore, the proposed algorithm is much faster and simpler to implement compared with the standard method. In addition, the robust constant modulus beamformer (RCMB) is also discussed as a special case of the QCMD beamformer. Simulation results demonstrate the efficiency of the gradient projection algorithm and superiority of the QCMD beamformer over several representative robust beamformers, indicating that it can approach the optimal performance bound.