Sound field decomposition using spherical microphone arrays

Abstract

Spherical microphone arrays offer a number of attractive properties such as direction-independent acoustic behavior and ability to reconstruct the sound eld in the vicinity of the array. Such ability is necessary in applications such as ambisonics and recreating auditory environment over headphones. We compare the performance of two scene reconstruction algorithms -one based on least-squares tting the observed potentials and another based on computing the far-eld signature function directly from the microphone measurements. A number of features important for the design and operation of spherical microphone arrays in real applications are revealed. Results indicate that it is possible to reconstruct the sound scene up to order p with p 2 microphones.Index Terms-Acoustic elds, spherical microphone arrays, array signal processing, acoustic position measurement.1. INTRODUCTION Spherical microphone arrays offer a number of properties attractive for the development of the acoustic and audio systems with 3-D listening capability. Due to 3-D symmetry of the array, the array beamforming pattern is independent of the steering direction and the spatial structure of the acoustic eld can be captured without distortion. [6] presented a framework for performing decomposition using spherical convolution under the assumption of a continuous pressure-sensitive microphone array surface. 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. BACKGROUNDIn 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,Thanks to the U.S. Department of Veterans Affairs for funding this work.where (k, r) is the Fourier transform of the pressure. 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]where (r, , ) are spherical coordinates of the radius vector r, j n (kr) and h n (kr) are the spherical Bessel and Hankel functions, and Y m n ( , ) are the orthonormal spherical harmonics. Any regular acoustic eld (k, r) near a point r in a region that does not contain sources can be represented as a sum of regular functions with some complex coef cients C m n (k) asTo achieve negligible truncation error it is suf cient to set [9]3. SOLVING THE ACOUSTIC SCENE The potential (s 0 , s) created at point s 0 on the surface of the soundhard sphere of radius a by plane wave e iks·r propagating in the direction s is given bywhere Pn(s 0 · s) is the Legendre polynomial of degree n an...

Plane-wave decomposition of a sound scene using a cylindrical microphone array

“…These cylindrical array results are consistent with similar simulations performed with spherical arrays in [10]. In particular, it is observed that the truncation number ps = p considered optimal in [9] is too high for use in real spherical and cylindrical beamforming applications as numerical problems arise even in a simulated setup.…”

“…The regularization parameter is related to the condition number of F . It was shown in [10] that the specific value of within reasonable bounds (from 10 7 to 10 2 relative to the elements of ) only marginally in uences the decomposition results.…”

“…An alternative algorithm for the spherical array case was introduced in [10]. It is based on finding the set of plane wave magnitudes μ(k, sj) that best explain the set of observed potentials (k, s i ).…”

“…Regularization is necessary because even though the computation of the elements of matrix F is not subject to numerical problems, the matrix itself can be ill-conditioned at lower frequencies and computation of the (pseudo)inverse of F can be unstable [10]. The regularization parameter is related to the condition number of F .…”

“…An alternative approach is to note that in the plane-wave basis [9] a sound field is represented as a weighted collection of plane waves arriving from various directions therefore, the mere act of converting a sound field into the plane-wave basis decomposes it into directional components. In a recent paper [10], this was done directly from the measurements at the microphones using a minimum least-squares (LS) procedure. It was shown that the LS method provides substantially the same beamforming and localization abilities while having a simpler mathematical formulation.…”