Efficient dual-sphere microphone-array design based on generalized sampling theory

The generalized sampling expansion, introduced by Papoulis. facilitates the reconstruction of a band- limited signal that has been sampled at a rate slower than the Nyquist rate - provided that additional information about the signal is available in the form of the sampled outputs of known linear systems with the original signal at their input. In this work, the generalized sampling expansion, originally- developed for time-domain signals, is formulated for functions over the sphere, using the spherical harmonics transform. The paper presents the theory of the new expansion that has been developed for the equal-angle sampling scheme. In the second part of the paper, the theory is applied to the design of an efficient dual-sphere microphone-array. A known problem, when sampling sound fields using an open-sphere microphone-array, is the ill-conditioning at specific frequencies due to the nulls of the spherical Bessel function. To overcome this problem, a dual-sphere design, which requires twice as many microphones compared to a single-sphere design, has previously been proposed. Applying the generalized sampling theory developed here, it is shown that a dual-sphere design with half the number of samples at each sphere can replace a single sphere, but only if the two spheres are rotated relative to each other in a specific manner. Reconstruction of the sound pressure on the sphere is then possible without increasing the total number of microphones, while at the same time countering the effect of the nulls.