Robust minimum variance beamforming

TLDR

Uncertainty in array response arises from imprecise angle‑of‑arrival knowledge and array‑manifold errors, and the method reduces to Capon's beamformer when the uncertainty ellipsoid collapses to a point. The paper extends minimum‑variance beamforming to explicitly account for uncertainty in array response. Uncertainty is modeled as an ellipsoid of possible array responses for a look direction, and weights are chosen to minimize weighted power while guaranteeing unit gain for all responses in the ellipsoid, a problem cast as a second‑order cone program solvable via Lagrange multipliers, with ellipsoids derived per signal‑path component and aggregated. New results on modeling element‑wise products of ellipsoids are presented, and numerical examples demonstrate the robust beamforming and ellipsoidal modeling methods.

Abstract

This paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response. Sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold. In our method, uncertainty in the array manifold is explicitly modeled via an ellipsoid that gives the possible values of the array for a particular look direction. We choose weights that minimize the total weighted power output of the array, subject to the constraint that the gain should exceed unity for all array responses in this ellipsoid. The robust weight selection process can be cast as a second-order cone program that can be solved efficiently using Lagrange multiplier techniques. If the ellipsoid reduces to a single point, the method coincides with Capon's method. We describe in detail several methods that can be used to derive an appropriate uncertainty ellipsoid for the array response. We form separate uncertainty ellipsoids for each component in the signal path (e.g., antenna, electronics) and then determine an aggregate uncertainty ellipsoid from these. We give new results for modeling the element-wise products of ellipsoids. We demonstrate the robust beamforming and the ellipsoidal modeling methods with several numerical examples.