Plane-wave decomposition analysis for spherical microphone arrays

Abstract

Spherical microphone arrays have attracted attention for analyzing the sound field in a region and beamforming. The analysis of the recorded sound has been performed in terms of spherical wavefunctions, and recently the use of plane-wave expansions has been suggested. We show that the plane-wave basis is intimately related to the spherical wave-functions. Reproduction in terms of both representations satisfies certain band-limit criteria. We provide an error bound that shows that to reproduce the spatial characteristics of a sound of a certain frequency we need to be able to accurately represent sounds of up to a particular order, which establishes a Nyquist-like criterion. The order of the sound field in turn is related to the number of microphones in the array necessary to achieve accurate quadrature on the sphere. These results are illustrated with simulations. INTRODUCTIONMany authors have proposed the use of spherical microphone arrays for beamforming and for capturing the spatial structure of the sound field in the region of space occupied by the array and in the nearby area. In [1] the sound field was captured using an open spherical microphone array. Microphones can also be positioned on the surface of a rigid sphere to make use of the scattering [2]. The design of the spherical arrays was considered in [3], and the analysis of the recorded sound field using plane-wave expansion was described in [4]. We present further details of the mathematical theory here, adding the effects of the finite number of microphones, the influence of the sound frequency, and error bounds. While plane-waves are often used to represent sound sources in the far-field, they also constitute a remarkable basis for the wave equation that can represent the sound field in both the near and farfields. This representation of the wave field is being exploited in developing the fast multipole method for the Helmholtz equation [5]. Using error bounds presented in [6], we build on all the papers mentioned above, show how analysis in terms of both wavefunctions and plane-waves can be used interchangeably, and provide explicit error bounds and frequency dependence. ANALYSIS OF THE SOUND FIELDSound propagation and sound scattering off the spherical array are governed by the wave equation (1, left). Using the Fourier transform (2, left), we can convert the wave equation to the Helmholtz equation in the frequency domain (1, right)where k is the wavenumber and ω = 2π f is the circular frequency. The wave propagation and scattering problem can be reduced then to solving the Helmholtz equation for a number of frequencies. To obtain the time domain solution, we take the inverse Fourier Transform (2, right) of ψ. For scattering problems the total (measured) field can be decomposed asIn this paper, we consider two problems: sound field analysis and beamforming. The goal of sound field analysis is to build an approximation to ψ in at the array location (center) using the sound recorded on the surface at the microphones, ψ, which is a combinat...

Sound field decomposition using spherical microphone arrays

“…In a space with no acoustic sources, acoustic wave propagation at a wavenumber k is governed by the Helmholtz equation [7] 2 (k, r) + k 2 (k, r) = 0,…”

“…Solutions of the Helmholtz equation can be expanded as a series of regular R m n (k, r) and singular S m n (k, r) spherical basis functions (see [7]): R …”

“…The goal is to measure the potentials (k, s 0 i ) at those microphones and to recover the signature function (k, sj) over Lj directions sj comprising an "S-grid". Two algorithms were introduced in [7]. K-method: This method is essentially based on least-squares tting of the observed potentials with the plane-wave magnitudes (sj).…”

“…Quadrature based on Fliege points [8] was presented and evaluated and two plane-wave decomposition algorithms were developed in [7], The current work analyzes the performance of those algorithms under realistic operating conditions -nite number of microphones, environmental noise, and aliasing effects -using both synthetic and experimental data.…”

“…In case of discrete microphones positioned on the sphere surface this assumption is invalid, and a quadrature formulae that preserves the orthonormality of spherical harmonics should be used as in [2]. Quadrature based on Fliege points [8] was presented and evaluated and two plane-wave decomposition algorithms were developed in [7], The current work analyzes the performance of those algorithms under realistic operating conditions -nite number of microphones, environmental noise, and aliasing effects -using both synthetic and experimental data. …”