Jørgen Hald

Sound source reconstruction using inverse boundary element calculations

Whereas standard boundary element calculations focus on the forward problem of computing the radiated acoustic field from a vibrating structure, the aim in this work is to reverse the process, i.e., to determine vibration from acoustic field data. This inverse problem is brought on a form suited for solution by means of an inverse boundary element method. Since the numerical treatment of the inverse source reconstruction results in a discrete ill-posed problem, regularization is imposed to avoid unstable solutions dominated by errors. In the present work the emphasis is on Tikhonov regularization and parameter-choice methods not requiring an error-norm estimate for choosing the right amount of regularization. Several parameter-choice strategies have been presented lately, but it still remains to be seen how well these can handle industrial applications with real measurement data. In the present work it is demonstrated that the L-curve criterion is robust with respect to the errors in a real measurement situation. In particular, it is shown that the L-curve criterion is superior to the more conventional generalized cross-validation (GCV) approach for the present tire noise studies.

Basic theory and properties of statistically optimized near-field acoustical holography

To avoid the requirement set by standard near-field acoustical holography (NAH) to measure an area that fully covers the source, a set of so-called patch NAH methods has been introduced. One such method is the statistically optimized NAH (SONAH). In this method, the acoustic quantities on a mapping surface near the measurement surface are calculated by using a transfer matrix defined in such a way that all propagating waves and a weighted set of evanescent waves are projected with optimal average accuracy. The present paper gives an overview of the basic theory of SONAH, including a description of phenomena such as spatial aliasing and wave-number domain leakage. A revised and generalized mathematical formulation is given, covering the calculation of all three components of particle velocity and the use of up to six virtual source planes. A set of formulas for the inherent estimation error level of the method is derived and used to visualize the regions of validity of the SONAH predictions for some typical microphone array geometries. The sensitivity of the inherent error level distribution to changes in the parameters of the SONAH algorithm is also investigated.

Source visualization by using statistically optimized near-field acoustical holography in cylindrical coordinates

Nearfield acoustical holography (NAH) is a useful tool for visualizing noise sources. However, to avoid spatial Fourier transform-related truncation effects, the measurement, or hologram, surface must extend beyond the source to a region where the sound pressure drops to a level significantly lower than the peak level within the measurement aperture. Statistically optimized nearfield acoustical holography (SONAH), first derived by Steiner and Hald in planar geometry, is based on a formulation similar to that of NAH. However, in SONAH, surface-to-surface projection of the sound field is performed by using a transfer matrix defined in such a way that all propagating waves and a weighted set of evanescent waves are projected with optimal average accuracy: i.e., no spatial Fourier transforms are performed. Thus the requirement that the measurement surface be extended is eliminated without compromising the accuracy of the procedure. In the present work, SONAH was re-formulated in cylindrical coordinates and wa...

Using beamforming and binaural synthesis for the psychoacoustical evaluation of target sources in noise

The potential of spherical-harmonics beamforming (SHB) techniques for the auralization of target sound sources in a background noise was investigated and contrasted with traditional head-related transfer function (HRTF)-based binaural synthesis. A scaling of SHB was theoretically derived to estimate the free-field pressure at the center of a spherical microphone array and verified by comparing simulated frequency response functions with directly measured ones. The results show that there is good agreement in the frequency range of interest. A listening experiment was conducted to evaluate the auralization method subjectively. A set of ten environmental and product sounds were processed for headphone presentation in three different ways: (1) binaural synthesis using dummy head measurements, (2) the same with background noise, and (3) SHB of the noisy condition in combination with binaural synthesis. Two levels of background noise (62, 72 dB SPL) were used and two independent groups of subjects (N=14) evaluated either the loudness or annoyance of the processed sounds. The results indicate that SHB almost entirely restored the loudness (or annoyance) of the target sounds to unmasked levels, even when presented with background noise, and thus may be a useful tool to psychoacoustically analyze composite sources.

Near field acoustic holography with microphones on a rigid sphere (L).

Spherical near field acoustic holography (spherical NAH) is a technique that makes it possible to reconstruct the sound field inside and just outside a spherical surface on which the sound pressure is measured with an array of microphones. This is potentially very useful for source identification. The sphere can be acoustically transparent or it can be rigid. A rigid sphere is somewhat more practical than an open sphere. However, spherical NAH based on a rigid sphere is only valid if it can be assumed that the sphere has a negligible influence on the incident sound field, and this is not necessarily a good assumption when the sphere is very close to a radiating surface. This Letter examines the matter through simulations and experiments.

Fast wideband acoustical holography

Patch near-field acoustical holography methods like statistically optimized near-field acoustical holography and equivalent source method are limited to relatively low frequencies, where the average array-element spacing is less than half of the acoustic wavelength, while beamforming provides useful resolution only at medium-to-high frequencies. With adequate array design, both methods can be used with the same array. But for holography to provide good low-frequency resolution, a small measurement distance is needed, whereas beamforming requires a larger distance to limit sidelobe issues. The wideband holography method of the present paper was developed to overcome that practical conflict. Only a single measurement is needed at a relatively short distance and a single result is obtained covering the full frequency range. The method uses the principles of compressed sensing: A sparse sound field representation is assumed with a chosen set of basis functions, a measurement is taken with an irregular array, and the inverse problem is solved with a method that enforces sparsity in the coefficient vector. Instead of using regularization based on the 1-norm of the coefficient vector, an iterative solution procedure is used that promotes sparsity. The iterative method is shown to provide very similar results in most cases and to be computationally much more efficient.

On the applicability of the spherical wave expansion with a single origin for near-field acoustical holography.

The spherical wave expansion with a single origin is sometimes used in connection with near-field acoustical holography to determine the sound field on the surface of a source. The radiated field is approximated by a truncated expansion, and the expansion coefficients are determined by matching the sound field model to the measured pressure close to the source. This problem is ill posed, and therefore regularization is required. The present paper investigates the consequence of using only the expansion truncation as regularization approach and compares it with results obtained when additional regularization (the truncated singular value decomposition) is introduced. Important differences between applying the method when using a microphone array surrounding the source completely and an array covering only a part of the source are described. Another relevant issue is the scaling of the wave functions. It is shown that it is important for the additional regularization to work properly that the wave functions are scaled in such a way that their magnitude on the measurement surface decreases with the order. Finally, the method is applied on nonspherical sources using a vibrating plate in both simulations and an experiment, and the performance is compared with the equivalent source method.

High-resolution Fly-over Beamforming Using a Small Practical Array

The paper describes a commercially available fly-over beamforming system based on methodologies already published, but using an array that was designed for quick and precise deployment on a concrete runway rather than for minimum sidelobe level. Time domain tracking Delay And Sum (DAS) beamforming is the first processing step, followed by Deconvolution in the frequency domain to reduce sidelobes, enhance resolution, and get absolute scaling of the source maps. The system has been used for a series of fly-over measurements on a Business Jet type MU300 from Mitsubishi Heavy Industries. Results from a couple of these measurements are presented: Contribution spectra from selected areas on the aircraft to the sound pressure level at the array are compared against the total sound pressure spectrum measured by the array. One major aim of the paper is to verify that the system performs well although the array was designed with quick deployment as a main criterion. The results are very encouraging. A second aim is to elaborate on the handling of the array shading function in connection with the calculation of the Point Spread Function (PSF) used in deconvolution. Recent publications have used a simple formula to compensate for Doppler effects for the case of flat broadband spectra. A more correct formula is derived in the present paper, covering also a Doppler correction to be made in the shading function, when that function is used in the PSF calculation.

Improving the efficiency of deconvolution algorithms for sound source localization

The localization of sound sources with delay-and-sum (DAS) beamforming is limited by a poor spatial resolution-particularly at low frequencies. Various methods based on deconvolution are examined to improve the resolution of the beamforming map, which can be modeled by a convolution of the unknown acoustic source distribution and the beamformers response to a point source, i.e., point-spread function. A significant limitation of deconvolution is, however, an additional computational effort compared to beamforming. In this paper, computationally efficient deconvolution algorithms are examined with computer simulations and experimental data. Specifically, the deconvolution problem is solved with a fast gradient projection method called Fast Iterative Shrikage-Thresholding Algorithm (FISTA), and compared with a Fourier-based non-negative least squares algorithm. The results indicate that FISTA tends to provide an improved spatial resolution and is up to 30% faster and more robust to noise. In the spirit of reproducible research, the source code is available online.

Panel Contribution Analysis in a Vehicle Cabin using a Dual Layer Handheld Array with Integrated Position Measurement

The paper describes an array-based method for measurement of panel contributions in a cabin. Both the FRF measurements and the operational measurements are taken with a dual layer microphone array, and the required sound field data on a panel surface mesh are calculated using the SONAH patch holography algorithm. Multiple array positions can be measured to cover the needed mesh area, and area-integration of contributions is then performed on the surface mesh. Measurement or alignment of surface mesh geometry and measurement of array positions is performed using an integrated 3D position measurement system. After a theoretical description the method is evaluated through simulated and actual measurements.