Eigenvalue beamforming using a multi-rank MVDR beamformer and subspace selection

Abstract—We derive eigenvalue beamformers to resolve an un-known signal of interest whose spatial signature lies in a known subspace, but whose orientation in that subspace is otherwise un-known. The unknown orientation may be fixed, in which case the signal covariance is rank-1, or it may be random, in which case the signal covariance is multirank. We present a systematic treat-ment of such signal models and explain their relevance for mod-eling signal uncertainties. We then present a multirank general-ization of the MVDR beamformer. The idea is to minimize the power at the output of a matrix beamformer, while enforcing a data dependent distortionless constraint in the signal subspace, which we design based on the type of signal we wish to resolve. We show that the eigenvalues of an error covariance matrix are funda-mental for resolving signals of interest. Signals with rank-1 covari-ances are resolved by the largest eigenvalues of the error covari-ance, while signals with multirank covariances are resolved by the smallest eigenvalues. Thus, the beamformers we design are eigen-value beamformers, which extract signal information from eigen-modes of an error covariance. We address the tradeoff between angular resolution of eigenvalue beamformers and the fraction of the signal power they capture. Index Terms—Eigenvalue beamforming, generalized sidelobe canceller, matched direction beamforming, matched subspace beamforming, multirank MVDR beamformer. I