Bayesian focusing for coherent wideband beamforming

In this paper, we present and study a Bayesian focusing transformation (BFT) for coherent wideband array processing, which takes into account the uncertainty of the direction of arrivals (DOAs). The Bayesian focusing method minimizes the mean-square error of the transformation over the probability density functions (pdfs) of the DOAs, thus achieving improved focusing accuracy over the entire bandwidth. In order to solve the Bayesian focusing problem, we derive and utilize a weighted extension of the wavefield interpolated narrowband generated subspace (WINGS) focusing transformation. We provide a closed-form expression for the optimal BFT and extend it to the case of directional sensors. We then consider a numerical computation scheme of the BFT in the angular domain. We show that if an angular sampling condition is satisfied then the angle domain approximation yields the optimal BFT. We also treat the important issue of robust focused minimum variance distortionless response (MVDR) beamformer. We analyze the sensitivity of the focused MVDR to the focusing errors and show that the array gain (AG) is inversely proportional to the square of the signal-to-noise ratio (SNR) for large values of the SNR, and highly sensitive to the focusing errors. In order to reduce this sensitivity we generalize the popular narrowband diagonal loaded MVDR to the focused wideband case, referred to as the Q-loaded focused MVDR wideband beamformer. We derive a closed-form analytic expression for the AG of the Q-loaded focused MVDR beamformer which depends on the focusing transformations. A numerical performance evaluation and simulations demonstrate the advantage of the BFT over that of other focusing transformations, for multiple source scenarios.