On Convergence of the Auxiliary-Vector Beamformer With Rank-Deficient Covariance Matrices

The auxiliary-vector beamformer is an algorithm that generates iteratively a sequence of beamformers which, under the assumption of a positive definite covariance matrix R, converges to the minimum variance distortionless response beamformer, without resorting to any matrix inversion. In the case where R is rank-deficient, e.g., when R is substituted for the sample covariance matrix and the number of snapshots is less than the number of array elements, the behavior of the AV beamformer is not known theoretically. In this letter, we derive a new convergence result and show that the AV beamformer weights converge when R is rank-deficient, and that the limit belongs to the class of reduced-rank beamformers..