Bayesian beamforming for DOA uncertainty: Theory and implementation

A Bayesian approach to adaptive narrowband beamforming for uncertain source direction-of-arrival (DOA) is presented. The DOA is modeled as a random variable with prior statistics that describe its level of uncertainty. The Bayesian beamformer is the corresponding minimum mean-square error (MMSE) estimator, which can be viewed as a mixture of directional beamformers combined according to the posterior distribution of the DOA given the data. Under a deterministic DOA, the mean-square error (MSE) of the Bayesian beamformer becomes as low as that of the directional beamformer equipped with the DOA candidate in the prior set that is the closest to the true DOA at exponential rate, where closeness is defined in the Kullback-Leibler sense. Two efficient algorithms using a uniform linear array (ULA) are presented. The first method utilizes the efficiency of the fast Fourier transform (FFT) to compute the posterior distribution on a large number of DOA candidates. The second method approximates the posterior distribution by a Gaussian distribution, which leads to a directional beamformer incorporated with a particular spreading matrix and an adjusted DOA. Numerical simulations show that the proposed beamformer outperforms other related blind or robust beamforming algorithms over a wide range of signal-to-noise ratios (SNRs).