Performance analysis of a dual microphone superdirective beamformer and approximate expressions for the near-field propagation regime

A linear array of sensors with small spacing (compared to the wavelength) can be processed with superdirective beamforming. Specifically when applying minimum variance distortionless response (MVDR) weights designed for a diffuse noise-field, high gains are attainable in theory. A classical result relating to the far-field regime states that the gain with respect to diffuse noise (i.e., the directivity factor) for a source in the endfire direction may approach the number of sensors squared (N²). However, as the wavelength increases, the beamformer encounters increasingly severe robustness issues. Results pertaining to the near-field regime are less well known. In this paper we analyze MVDR beamforming in a generic dual-microphone array scenario. Our analysis is not restricted to the far-field regime. We derive precise expressions for the directivity factor and the white-noise gain, as well as simplified approximations for the near- and far-field regimes. We show that in the near-field regime the directivity factor approaches infinity as the wavelength increases, and that the white-noise gain depends only on the ratio between the distance from the source to the distance between sensors. These properties of the beamformer (BF) behave differently than in the far-field regime.