Eigenvalue Beamforming using a Multi-rank MVDR Beamformer

We derive multi-rank generalizations of the MVDR beamformer to separate an unknown signal of interest in the presence of interference and noise. The spatial signature of the signal is assumed to lie in a known linear subspace, but the orientation of the signal in that subspace is otherwise unknown. The unknown orientation may be fixed for a sequence of experimental realizations, in which case the signal has a rank1 covariance matrix, or it may be random over realizations, in which case the signal covariance is multi-rank. We derive two generalizations of the MVDR beamformer: a matched direction beamformer, for the rank-1 case where the signal is drawn from an unknown but fixed direction within a known multidimensional subspace, and a matched subspace beamformer, for the multi-rank case where the signal is drawn from a known multi-dimensional subspace.