A comparison of compressive equivalent source methods for distributed sources.

Sparse sound-field reconstruction methods based on compressive sensing and the equivalent source method have gained a lot of interest in recent years, offering a wide frequency range. An irregular array must be employed, and the sound field must be representable by a sparse vector of source-model amplitudes. With few and concentrated physical sources, the sparsity assumption can be fulfilled, but distributed sources cause problems. Several methods have therefore been introduced to support sparse representation of distributed sources. One set of such methods represents the amplitude vector as a linear combination of modal amplitude distributions. In that case, the coefficient vector just has to be sparse. The present paper gives an overview of such methods, and the performance of the methods is compared based on a set of simulated measurements. Overall the modal representations work well if the true source distribution can be represented by relatively few modes. The results show that even for a vibrating plate this may not be the case if the source model size does not match the plate size and/or if a non-central plate excitation is applied. If, in addition, there are also compact sources, then the basic method without modes may be the best choice.

Denoising of cross-spectral matrices using canonical coherence.

Disturbing noise in microphone signals is an often occurring problem. The present paper has a main focus on measurements with microphone arrays in wind tunnels, where flow noise will be generated in the individual microphones. Considering stationary operational conditions, the cross-spectral matrix (CSM) is first averaged. Subsequently it is applied for aerodynamic noise-source mapping on a vehicle in the flow using one of several possible array processing algorithms. Several methods exist to suppress the flow-noise effects. Long-time averaging will gradually concentrate the flow-noise contributions on the CSM diagonal, and some array processing algorithms can avoid use of the diagonal. Other algorithms need the diagonal. Denoising methods that subtract a maximum of signal power from the diagonal, while retaining all off-diagonal elements unchanged and the matrix positive semidefinite, have been shown to be limited by remaining off-diagonal flow-noise contributions. A couple of algorithms exist that can overcome that limitation by supporting modifications outside the diagonal, but these methods seem to be limited by high computational cost or difficult parameter selection. The present paper introduces a unique canonical-coherence-based method, which is computationally very fast and with automated parameter selection. The performance is investigated through simulated and real measurements.

A comparison of iterative sparse equivalent source methods for near-field acoustical holography.

During the past decade, several publications have described the use of compressive sensing principles to extend the frequency range supported by a given irregular microphone array for near-field acoustic holography. The applied numerical source model has typically been of the type used for the equivalent source method, i.e., a mesh of point sources, and a one-norm regularized inverse problem has been solved using a very stable, but slow interior-point optimization algorithm. A few publications have investigated the use of simpler and faster iterative algorithms. The present paper gives a brief description of five such iterative algorithms, and it compares their performances with that of the interior-point algorithm based on a set of simulated measurements. A particular focus is on the suitability for industrial applications. Finally, an optimal choice of methodology is discussed based on the presented limited set of simulated tests.

Removal of incoherent noise from an averaged cross-spectral matrix.

Measured cross-spectral matrices (CSMs) from a microphone array will in some cases be contaminated by severe incoherent noise signals in the individual channels. A typical example is flow noise generated in the individual microphones when measuring in a wind tunnel. Assuming stationary signals and performing long-time averaging, the contamination will be concentrated on the CSM diagonal. When the CSM is used for traditional frequency-domain beamforming, diagonal removal (DR) will avoid use of the diagonal. DR is effective at suppressing the contamination effects, but it also has some side effects. With other beamforming algorithms and in connection with acoustic holography, however, the diagonal of the CSM is needed. The present paper describes a method for removal of incoherent noise contamination from the CSM diagonal. The method formulates the problem as a semidefinite program, which is a convex optimization problem that can be solved very efficiently and with guaranteed convergence properties. A first numerical study investigates the question, whether the semidefinite program formulation will provide in all cases the desired output. A second numerical study investigates the limitations introduced by off-diagonal noise contributions due to finite averaging time. The results of that study are backed up by results from a practical measurement.

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.

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 beamformer's 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.

Scaling of plane-wave functions in statistically optimized near-field acoustic holography.

Statistically Optimized Near-field Acoustic Holography (SONAH) is a Patch Holography method, meaning that it can be applied in cases where the measurement area covers only part of the source surface. The method performs projections directly in the spatial domain, avoiding the use of spatial discrete Fourier transforms and the associated errors. First, an inverse problem is solved using regularization. For each calculation point a multiplication must then be performed with two transfer vectors--one to get the sound pressure and the other to get the particle velocity. Considering SONAH based on sound pressure measurements, existing derivations consider only pressure reconstruction when setting up the inverse problem, so the evanescent wave amplification associated with the calculation of particle velocity is not taken into account in the regularized solution of the inverse problem. The present paper introduces a scaling of the applied plane wave functions that takes the amplification into account, and it is shown that the previously published virtual source-plane retraction has almost the same effect. The effectiveness of the different solutions is verified through a set of simulated measurements.

Psychoacoustic evaluation of multichannel reproduced sounds using binaural synthesis and spherical beamforming.

The binaural auralization of a 3D sound field using spherical-harmonics beamforming (SHB) techniques was investigated and compared with the traditional method using a head-and-torso simulator (HATS). The new procedure was verified by comparing simulated room impulse responses with measured ones binaurally. The objective comparisons show that there is good agreement in the frequency range between 0.1 and 6.4 kHz. A listening experiment was performed to validate the SHB method subjectively and to compare it to the HATS method. Two musical excerpts, pop and classical, were used. Subjective responses were collected in two head rotation conditions (fixed and rotating) and six spatial reproduction modes, including phantom mono, stereo, and surround sound. The results show that subjective scales of width, spaciousness, and preference based on the SHB method were similar to those obtained for the HATS method, although the width and spaciousness of the stimuli processed by the SHB method were judged slightly higher than the ones using the HATS method in general. Thus, binaural synthesis using SHB may be a useful tool to reproduce a 3D sound field binaurally, while saving considerably on measurement time because head rotation can be simulated based on a single recording.

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.

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.

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.

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.

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.