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Beamforming in the Reproducing Kernel Domain Based on Spatial Differentiation

Takahiro Iwami, Naohisa Inoue, Akira Omoto

TL;DR

The paper develops a beamforming framework in the reproducing-kernel domain by treating directional reception as spatial differentiation using polynomial differential operators. This RK-domain approach unifies axisymmetric and non-axisymmetric beam patterns and connects to spherical-harmonic beamforming via Hobson’s theorem, enabling explicit control of directivity and placement in interior fields. It introduces a reproducing-kernel representation with kernel $\kappa_{k}(\mathbf{r},\mathbf{r}')=\mathrm{J}_{d,0}(k|\mathbf{r}-\mathbf{r}'|)$ and derives practical estimators and weights for simple and general beamformers, including lifting axisymmetric SH formulations to arbitrary directions and positions. Numerical simulations in 2D validate sound-field reconstruction, beamforming DI, and directional-field extraction, demonstrating improved performance over omnidirectional models and revealing the framework’s flexibility and scalability. The work provides a theoretically grounded, extensible toolset for high-resolution, position-aware beamforming in arbitrary dimensions with explicitly modeled microphone directivities.

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.

Beamforming in the Reproducing Kernel Domain Based on Spatial Differentiation

TL;DR

The paper develops a beamforming framework in the reproducing-kernel domain by treating directional reception as spatial differentiation using polynomial differential operators. This RK-domain approach unifies axisymmetric and non-axisymmetric beam patterns and connects to spherical-harmonic beamforming via Hobson’s theorem, enabling explicit control of directivity and placement in interior fields. It introduces a reproducing-kernel representation with kernel and derives practical estimators and weights for simple and general beamformers, including lifting axisymmetric SH formulations to arbitrary directions and positions. Numerical simulations in 2D validate sound-field reconstruction, beamforming DI, and directional-field extraction, demonstrating improved performance over omnidirectional models and revealing the framework’s flexibility and scalability. The work provides a theoretically grounded, extensible toolset for high-resolution, position-aware beamforming in arbitrary dimensions with explicitly modeled microphone directivities.

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.

Paper Structure

This paper contains 21 sections, 2 theorems, 71 equations, 8 figures, 2 tables.

Table of Contents

  1. INTRODUCTION
  2. Notation and Preliminaries
  3. Interpretation of Directional Reception and Sound Field Reconstruction
  4. Directional Reception Model Based on Polynomial Differential Operators
  5. Sound Field Reconstruction
  6. Beamformers in the RK Domain
  7. Simple beamformer
  8. General beamformer
  9. Extraction of Directional Sound Fields by Beamforming
  10. Extraction of Directional Sound Fields by Beamforming
  11. Interpretation of Axisymmetric Beamformers in the Spherical Harmonic Domain
  12. Axisymmetric Beamformers in the Spherical Harmonic Domain
  13. Interpretation via Spatial Differential Operators
  14. Numerical Simulations
  15. Sound Field Reconstruction Performance
  16. ...and 6 more sections

Key Result

Proposition 1

(Hobson's formula) Assume that $f \in C^{\infty}(\mathbb{R}^{d})$ can be regarded as a radial function $f(\bm{r}) = f_{0}(|\bm{r}|)$. Then, for any $g \in \mathscr{P}_{\nu}$, the following relation holds:

Figures (8)

  • Figure 1: (Color online) Microphone configuration used in the numerical simulations.
  • Figure 2: (Color online) MNEs of the existing and proposed methods.
  • Figure 3: (Color online) Reference and reconstructed sound fields at 100 Hz and 1000 Hz.
  • Figure 4: (Color online) Directivity indices (DIs) of the proposed method.
  • Figure 5: (Color online) Beam patterns of the proposed method at 250, 500, 1000, 2000, and 4000 Hz.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Proposition 1
  • Corollary 1