We derive eigenvalue beamformers to resolve an unknown signal of interest whose spatial signature lies in a known subspace, but whose orientation in that subspace is otherwise unknown. 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 treatment of such signal models and explain their relevance for modeling signal uncertainties. We then present a multirank generalization 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 fundamental for resolving signals of interest. Signals with rank-1 covariances are resolved by the largest eigenvalues of the error covariance, while signals with multirank covariances are resolved by the smallest eigenvalues. Thus, the beamformers we design are eigenvalue beamformers, which extract signal information from eigenmodes of an error covariance. We address the tradeoff between angular resolution of eigenvalue beamformers and the fraction of the signal power they capture.
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