Beamforming in the Reproducing Kernel Domain Based on Spatial Differentiation

TL;DR

This paper introduces a new beamforming approach in the reproducing kernel domain that uses spatial differentiation to create flexible, arbitrary beam patterns, enhancing the theoretical understanding and extensibility of existing methods.

Contribution

It develops a unified kernel-based beamforming framework using polynomial differential operators, generalizing and clarifying spherical harmonic domain beamformers.

Findings

Validated through numerical simulations in 2D space

Able to generate arbitrary beam patterns including non-axisymmetric

Provides analytical expressions supported by Hobson's theorem

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.