Robust Beamforming by Linear Programming

In this paper, a robust linear programming beamformer (RLPB) is proposed for non-Gaussian signals in the presence of steering vector uncertainties. Unlike most of the existing beamforming techniques based on the minimum variance criterion, the proposed RLPB minimizes the $\ell_{\infty}$-norm of the output to exploit the non-Gaussianity. We make use of a new definition of the $\ell_{p}$-norm $(1\leq p\leq\infty)$ of a complex-valued vector, which is based on the $\ell_{p}$-modulus of complex numbers. To achieve robustness against steering vector mismatch, the proposed method constrains the $\ell_{\infty}$-modulus of the response of any steering vector within a specified uncertainty set to exceed unity. The uncertainty set is modeled as a rhombus, which differs from the spherical or ellipsoidal uncertainty region widely adopted in the literature. The resulting optimization problem is cast as a linear programming and hence can be solved efficiently. The proposed RLPB is computationally simpler than its robust counterparts requiring solution to a second-order cone programming. We also address the issue of appropriately choosing the uncertainty region size. Simulation results demonstrate the superiority of the proposed RLPB over several state-of-the-art robust beamformers and show that its performance can approach the optimal performance bounds.