Wolfgang Kreuzer

Fast multipole boundary element method to calculate head-related transfer functions for a wide frequency range

Head-related transfer functions (HRTFs) play an important role in spatial sound localization. The boundary element method (BEM) can be applied to calculate HRTFs from non-contact visual scans. Because of high computational complexity, HRTF simulations with BEM for the whole head and pinnae have only been performed for frequencies below 10 kHz. In this study, the fast multipole method (FMM) is coupled with BEM to simulate HRTFs for a wide frequency range. The basic approach of the FMM and its implementation are described. A mesh with over 70 000 elements was used to calculate HRTFs for one subject. With this mesh, the method allowed to calculate HRTFs for frequencies up to 35 kHz. Comparison to acoustically-measured HRTFs has been performed for frequencies up to 16 kHz, showing a good congruence below 7 kHz. Simulations with an additional shoulder mesh improved the congruence in the vertical direction. Reduction in the mesh size by 5% resulted in a substantially-worse representation of spectral cues. The effects of temperature and mesh perturbation were negligible. The FMM appears to be a promising approach for HRTF simulations. Further limitations and potential advantages of the FMM-coupled BEM are discussed.

A time-frequency method for increasing the signal-to-noise ratio in system identification with exponential sweeps

Exponential sweeps are widely used to measure impulse responses of electro-acoustic systems. Measurements are often contaminated by environmental noise and nonlinear distortions. We propose a method to increase the signal-to-noise ratio (SNR) by denoising the recorded signal in the time-frequency plane. In contrast to state-of-the art denoising methods, no assumption about the spectral characteristics of the noise is required. Numerical simulations demonstrate improvements in the SNR under low-SNR conditions even for measurements contaminated by colored noise.

A FORMULATION OF THE FAST MULTIPOLE BOUNDARY ELEMENT METHOD (FMBEM) FOR ACOUSTIC RADIATION AND SCATTERING FROM THREE-DIMENSIONAL STRUCTURES

Compared to the traditional boundary element method (BEM), the single level fast multipole boundary element method (SLFMBEM) or the multilevel fast multipole boundary element method (MLFMBEM) reduces the computational complexity of a job from O(n2) to O(n3/2) or O(n log2n), respectively with n being the number of unknowns; this means a dramatical reduction in terms of CPU-time and storage requirement. Large scale problems, unsolvable with the traditional BEM, can be solved by using the FMBEM. In this paper, the traditional BEM, SLFMBEM, and MLFMBEM are formulated within the framework of the Burton–Miller Collocation BEM for acoustic radiation and scattering from 3D structures. Attention is especially paid to the practical aspects of the method in order to get a reliable and efficient computation code. The performance of the method is tested with practical examples, including one for computing the head-related transfer function (HRTF) between 1000 and 18 000 Hz.

Numerical calculation of listener-specific head-related transfer functions and sound localization: Microphone model and mesh discretization

Head-related transfer functions (HRTFs) can be numerically calculated by applying the boundary element method on the geometry of a listeners head and pinnae. The calculation results are defined by geometrical, numerical, and acoustical parameters like the microphone used in acoustic measurements. The scope of this study was to estimate requirements on the size and position of the microphone model and on the discretization of the boundary geometry as triangular polygon mesh for accurate sound localization. The evaluation involved the analysis of localization errors predicted by a sagittal-plane localization model, the comparison of equivalent head radii estimated by a time-of-arrival model, and the analysis of actual localization errors obtained in a sound-localization experiment. While the average edge length (AEL) of the mesh had a negligible effect on localization performance in the lateral dimension, the localization performance in sagittal planes, however, degraded for larger AELs with the geometrical error as dominant factor. A microphone position at an arbitrary position at the entrance of the ear canal, a microphone size of 1 mm radius, and a mesh with 1 mm AEL yielded a localization performance similar to or better than observed with acoustically measured HRTFs.

Design of Spatial Microphone Arrays for Sound Field Interpolation

This paper presents a design method for microphone arrays with arbitrary geometries. Based on a theoretical analysis and on the magic points method, it allows for the interpolation of a sound field in a generic convex domain with a limited number of microphones on a given frequency band. It is shown that only a few microphones are needed in the interior of the considered domain to ensure a low interpolation error in the frequency band of interest, and that most of the microphones have to be located on the boundary of the domain, with a non-uniform density depending on the shape of the domain. Practical design constraints can be included in the optimization process. Comparisons for some particular array geometries with design methods known from the literature are given, showing that the proposed approach results in lower errors.

A STOCHASTIC 2D-MODEL FOR CALCULATING VIBRATIONS IN RANDOM LAYERS

Vibrations induced by machinery and traffic have become of increasing concern in the last years, for example, when constructing buildings near railway lines. In this paper we will present a model designed to predict the vibration level in the ground. Since in practice it is nearly impossible to determine exact material parameters for soil layers, we use a model with a stochastic shear modulus G. Under moderate assumptions G can be split with the Karhunen–Loeve expansion into a mean value G0 and a stochastic part Gstoch. Using a combination method of finite elements, Fourier transformation and Polynomial Chaos, it is possible to transform the partial differential equation describing the system into a matrix-vector formulation Kx = b which can be split into a deterministic and a stochastic part (K0 + Ks) x = b analog to the shear modulus. To keep the dimensions of the matrices involved with this system small, we use a Neumann-like iteration to solve it. Finally, results for a small example are presented.

Design of a robust open spherical microphone array

This paper presents a method for designing a robust open spherical microphone array that overcomes the typical problems of open sphere geometries at frequencies related to the zeros of the spherical Bessel functions. The proposed array structure uses only a few additional sampling points inside the spherical volume whose optimal positions are determined by the eigenmodes of the sphere for a given wave number interval. This novel approach minimizes the interpolation error inside the sphere. With illustrate this approach with the design of a 10-th order array using 130 microphones and discuss the simulation results with regard to commonly used error measures (white noise gain, condition number, and interpolation error), and show that the proposed array design compares favorably to previously suggested array designs.

A 3D MODEL TO SIMULATE VIBRATIONS IN A LAYERED MEDIUM WITH STOCHASTIC MATERIAL PARAMETERS

A major problem for the simulation of the propagation of vibrations in ground layers is the fact that it is almost impossible to determine the material parameters needed for a numerical model exactly. In this work, we present a 3D model for layered soil, where in each layer the shear modulus is modeled as a stochastic process. Using the Karhunen Loeve expansion, the polynomial chaos expansion, and the Fourier transform, the stochastic system can be transformed into a linear system of equations in the wavenumber frequency domain. Unfortunately, the size of this system becomes very large and — contrary to a deterministic system — the stochastic system can no longer be decoupled for every wavenumber in the spatial Fourier domain. To solve this system efficiently, we propose an iteration procedure where the system is split into a deterministic and a stochastic part. As an external load on top of the ground, we use a vibrating box load moving along the x-axis. We discuss implementational details and present simulation results.

A priori mesh grading for the numerical calculation of the head-related transfer functions

Head-related transfer functions (HRTFs) describe the directional filtering of the incoming sound caused by the morphology of a listeners head and pinnae. When an accurate model of a listeners morphology exists, HRTFs can be calculated numerically with the boundary element method (BEM). However, the general recommendation to model the head and pinnae with at least six elements per wavelength renders the BEM as a time-consuming procedure when calculating HRTFs for the full audible frequency range. In this study, a mesh preprocessing algorithm is proposed, viz., a priori mesh grading, which reduces the computational costs in the HRTF calculation process significantly. The mesh grading algorithm deliberately violates the recommendation of at least six elements per wavelength in certain regions of the head and pinnae and varies the size of elements gradually according to an a priori defined grading function. The evaluation of the algorithm involved HRTFs calculated for various geometric objects including meshes of three human listeners and various grading functions. The numerical accuracy and the predicted sound-localization performance of calculated HRTFs were analyzed. A-priori mesh grading appeared to be suitable for the numerical calculation of HRTFs in the full audible frequency range and outperformed uniform meshes in terms of numerical errors, perception based predictions of sound-localization performance, and computational costs.