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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.

Explicit expressions are developed for the cross-spectral density between pairs of sensors and the wavenumber-frequency spectrum projected onto the line joining these sensors, when they are placed in arbitrary positions in noise fields which are described by an arbitrary directional distribution of uncorrelated plane waves. The approach is to expand the directional distribution in spatial harmonics. Each harmonic leads to a corresponding term, with the same coefficient, in series representations for the cross-spectral density function and the wave-number-frequency spectrum. The approach particularly attractive when the field can be adquately represented by a relatively small number of harmonics. Both two- and three-dimensional fields are considered. The approach is applied to the vertical directionality of ambient sea noise and is related to some existing models of ambient sea noise. Some new models are presented. Results are compared with reported experimental data and found to be in good agreement over a wide frequency range.