Array pattern synthesis with robustness against manifold vectors uncertainty

The directivity pattern of an array is known to degrade in the presence of errors in the array manifolds, with respect to the desired nominal array pattern. This paper describes a new robust pattern synthesis approach to arrays with manifold vectors perturbation. This synthesis technique optimizes the worst case performance by minimizing the worst case sidelobe level while maintaining a distortionless respect to the worst case signal steering vector. The possible values of the manifold are covered by an ellipsoid that describes the uncertainty in terms of errors in element gains and phase angles. The pattern synthesis parameters can be optimally chosen based on known levels of uncertainty in the manifold vectors. Two optimization criteria, l₁ regularization and l₂ regularization, of a robust beamformer are proposed. Both criteria of the robust beamformer problem can be reformulated in a convex form of second-order cone programming, which is computationally tractable. A simple lower bound on the difference between the worse case sidelobe level of the robust beamformer and the sidelobe level of the nominal optimal beamformer with no array manifold uncertainty is derived. This robust approach is applicable to arrays with arbitrary geometry. Its effectiveness is illustrated through its application to a circular hydrophone array. An experiment is performed to measure the manifold vectors uncertainty set of hydrophone arrays. Results of applying the algorithms to both simulated and experimental data are presented and they show good performance of the proposed robust pattern synthesis approach.