Resolving power and sensitivity to mismatch of optimum array processors

Abstract

Mismatch in a beamformer occurs when the knowledge of the signal directional properties is imprecise. The effects of mismatch on a conventional beamformer and two optimum beamformers are compared. One optimum beamformer is based on inversion of the noise cross-spectral matrix while the other is based on inversion of the signal-plus-noise cross-spectral matrix. When there is mismatch, the inclusion of the signal in the matrix inversion process can lead to dramatic reductions in the output signal-to-noise ratio when the output signal-to-noise ratio of a perfectly matched beamformer would be greater than unity. However, the corresponding effect on the total beamformer output is less dramatic since an increase in the noise response partially offsets the decrease in signal response. The question of suppressing mismatched signals is closely related to the question of resolving closely spaced sources. Exact conditions are presented for resolution of closely spaced sources by an optimum beamformer. These results are applied to a line array and compared with the resolution capability of a conventional beamformer. It is found, for example, that an output signal-to-noise ratio of about 47 dB is required to achieve a resolving power with an optimum processor which is ten times that given by the classical Rayleigh limit. Conditions are also presented for the resolution of two sources of unequal strength.