Beamforming in the Reproducing Kernel Domain Based on Spatial Differentiation

Abstract

This paper proposes a novel beamforming framework in the reproducing kernel domain, derived from a unified interpretation of directional response as spatial differentiation of the sound field. By representing directional response using polynomial differential operators, the proposed method enables the formulation of arbitrary beam patterns including non-axisymmetric. The derivation of the reproducing kernel associated with the interior fields is mathematically supported by Hobson's theorem, which allows concise analytical expressions. Furthermore, the proposed framework generalizes conventional spherical harmonic domain beamformers by reinterpreting them as spatial differential operators, thereby clarifying their theoretical structure and extensibility. Three numerical simulations conducted in two-dimensional space confirm the validity of the method.