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Area of Science:

  • Acoustics
  • Signal Processing
  • Computational Mathematics

Background:

  • Conventional beamforming methods often rely on specific mathematical domains like spherical harmonics (SH).
  • Representing directional responses and arbitrary beam patterns, especially non-axisymmetric ones, presents a significant challenge.
  • A unified framework for interpreting directional response as spatial differentiation could offer greater flexibility.

Purpose of the Study:

  • To propose a novel beamforming framework in the reproducing kernel domain.
  • To enable the formulation of arbitrary and non-axisymmetric beam patterns.
  • To generalize and clarify the theoretical structure of existing spherical harmonic (SH) domain beamformers.

Main Methods:

  • Interpreting directional response as spatial differentiation of the sound field.
  • Representing directional response using polynomial differential operators.
  • Reinterpreting beamforming as a functional parameterized by directivity function and virtual observation position.
  • Deriving the reproducing kernel for interior fields using Hobson's theorem.

Main Results:

  • The proposed framework allows for arbitrary beam pattern formulation, including non-axisymmetric ones.
  • Mathematical derivation using Hobson's theorem provides concise analytical expressions.
  • The framework generalizes conventional SH domain beamformers, clarifying their theoretical underpinnings.
  • Numerical simulations in 2D and 3D confirm the method's validity.

Conclusions:

  • The reproducing kernel domain beamforming framework offers a flexible and unified approach.
  • Explicitly modeling microphone directivity enhances sound field reconstruction accuracy.
  • The proposed method achieves performance comparable to SH domain beamforming, with slight advantages at higher frequencies.