Uniformly robust mean-squared error beamforming

We consider the problem of designing a linear beamformer to estimate a source signal s(t) from array observations, where the goal is to obtain an estimate ŝ(t) that is close to s(t). Although standard beamforming approaches are aimed at maximizing the signal-to-interference-plus-noise ratio (SINR), maximizing SINR does not guarantee a small mean-squared error (MSE), hence on average a signal estimate maximizing the SINR can be far from s(t). To ensure that ŝ(t) is close to s(t), we propose using the more appropriate design criterion of MSE. Since the MSE depends in general on s(t) which is unknown, it cannot be minimized directly. Instead, we suggest two beamforming methods that minimize a worst-case measure of MSE. We first consider a minimax MSE beamformer that minimizes the worst-case MSE. We then consider a minimax regret beamformer that minimizes the worst-case difference between the MSE using a beamformer ignorant of s(t) and the smallest possible MSE attainable with a beamformer that knows s(t). We demonstrate through numerical examples that the proposed minimax methods outperform several existing standard and robust beamformers, over a wide range of SNR values.