Spherical Fraction Beamforming

This paper describes a beamforming method for one-eighth, quarter, and half-spaces, bounded by rigid planes. The proposed approach is based on the spherical fraction harmonic decomposition, similar to that of spherical harmonics decomposition for the whole sphere. The definition of these functions is given in detail. It is also proved that they form a sub-basis of the spherical harmonics, with a fraction-dependent normalization gain, which makes it easy to adapt from existing spherical harmonics implementation. Taking advantage of the rigid boundary condition, it is shown that the number of spherical fraction harmonics used to construct a beamformer is less than the number of spherical harmonics used in the case of an unbounded domain. However, image beams are generated and can degrade the performance of the beamformer. For the response to a plane wave, it is shown that the angular error on the maximum, as well as the directivity factor are a function of the spherical fraction under consideration, the plane wave direction of arrival and the degree of decomposition $L$. As the latter increases and the spherical fraction is large, these two parameters are converging towards the unbounded case values with fewer functions to evaluate. An application case is presented.