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Frequency-Domain Sound Field from the Perspective of Band-Limited Functions

Takahiro Iwami, Akira Omoto

TL;DR

This work addresses representing the frequency-domain sound field in general dimension as a band-limited function whose wavenumber spectrum is confined to the sphere. It builds an RKHS with a kernel defined by an angular integral and uses kernel ridge regression to reconstruct the field from samples, yielding a practical estimator. Wavenumber-spectrum estimation is achieved by applying a spatial Fourier transform to the kernel, enabling directional information from the forward-wave component. Numerical results in 2D validate improved field reconstruction over a spherical-harmonic baseline and show plausible, direction-aware spectra, supporting sparse-sensor deployment and potential DOA estimation.

Abstract

In this paper, the frequency-domain sound field is regarded as an element of some band-limited function space, and a representation of the field as a linear combination of the reproducing kernel in that space is proposed. This model has the strongest representational capacity of all function systems when we know only the sound pressure information at arbitrary positions. The proposed model can be considered a generalization of the existing three-dimensional sound field model using the reproducing kernel of the solution space of the Helmholtz equation to the spatial dimension. One of the advantages of capturing the frequency-domain sound field in this way is the simplicity achieved for the estimation formula of the wavenumber spectrum. Two numerical simulations were conducted to validate the proposed methods.

Frequency-Domain Sound Field from the Perspective of Band-Limited Functions

TL;DR

This work addresses representing the frequency-domain sound field in general dimension as a band-limited function whose wavenumber spectrum is confined to the sphere. It builds an RKHS with a kernel defined by an angular integral and uses kernel ridge regression to reconstruct the field from samples, yielding a practical estimator. Wavenumber-spectrum estimation is achieved by applying a spatial Fourier transform to the kernel, enabling directional information from the forward-wave component. Numerical results in 2D validate improved field reconstruction over a spherical-harmonic baseline and show plausible, direction-aware spectra, supporting sparse-sensor deployment and potential DOA estimation.

Abstract

In this paper, the frequency-domain sound field is regarded as an element of some band-limited function space, and a representation of the field as a linear combination of the reproducing kernel in that space is proposed. This model has the strongest representational capacity of all function systems when we know only the sound pressure information at arbitrary positions. The proposed model can be considered a generalization of the existing three-dimensional sound field model using the reproducing kernel of the solution space of the Helmholtz equation to the spatial dimension. One of the advantages of capturing the frequency-domain sound field in this way is the simplicity achieved for the estimation formula of the wavenumber spectrum. Two numerical simulations were conducted to validate the proposed methods.
Paper Structure (12 sections, 43 equations, 4 figures, 2 tables)

This paper contains 12 sections, 43 equations, 4 figures, 2 tables.

Table of Contents

  1. INTRODUCTION
  2. Space of band-limited functions to which frequency-domain sound fields belong and its RK
  3. Sound field expression and wavenumber spectrum estimation
  4. Sound field reconstruction
  5. Estimation of the wavenumber spectrum
  6. Comparison of models between the instantaneous sound field and frequency-domain sound field
  7. Numerical simulation
  8. Comparison with the existing method
  9. Wavenumber spectrum estimation
  10. Concluding remarks
  11. Proof that the wavenumber spectral support of the sound field is restricted on a unit sphere
  12. Derivation of the RK

Figures (4)

  • Figure 1: Visualization of the wavenumber spectrum for the existing model and proposed model. Whereas the existing model can have the entire sphere as its support, the proposed model is restricted to the surface of the sphere.
  • Figure 2: Problem setting in Sec. \ref{['sec:3']}.
  • Figure 3: Estimated sound fields and their NE distributions when using the existing and proposed models.
  • Figure 4: Result of the proposed estimation of the wavenumber spectrum.