Least squares frequency-invariant beamforming

Frequency invariant beamforming aims to parametrize array filter coefficients such that the spectral and spatial response profiles of the array can be adjusted independently. Solutions to this problem have been presented for particular array geometries and rely often on inversion formulas for Fourier or spherical harmonics. These decompositions are analytically appealing but require a larger number of sensors and/or a regular microphone spacing. However, in practical applications the number and location of sensors are often restricted. This paper proposes to use a linear basis that optimally reproduces a desired spatial response pattern for each frequency. This numerical least-squares inversion is applicable to arbitrary sensor configurations for which typically no exact analytical inverses are available. This basis can be combined further with spherical harmonics resulting in a readily steerable and low dimensional parametrization. This solution to frequency invariant beamforming effectively decouples the array geometry from the steering geometry. Here the method is demonstrated for the optimal design of the far-field response of an irregular linear array with as few as 3 microphones combined with Legendre polynomials to control the azimuth orientation of the frequency-invariant beam