Gain and phase calibration of sensor arrays from ambient noise by cross-spectral measurements fitting.

We address the problem of blind gain and phase calibration of a sensor array from ambient noise. The key motivation is to ease the calibration process by avoiding a complex procedure setup. We show that computing the sample covariance matrix in a diffuse field is sufficient to recover the complex gains. To do so, we formulate a non-convex least-square problem based on sample and model covariances. We propose to obtain a solution by low-rank matrix approximation, and two efficient proximal algorithms are derived accordingly. The first algorithm solves the problem modified with a convex relaxation to guarantee that the solution is a global minimizer, and the second algorithm directly solves the initial non-convex problem. We investigate the efficiency of the proposed algorithms by numerical and experimental results according to different sensing configurations. These results show that efficient calibration highly depends on how the measurements are correlated. That is, estimation is achieved more accurately when the field is spatially over-sampled.

Galerkin equivalent sources method for sound field reconstruction around diffracting bodies.

The rising interest for three-dimensional acoustic imaging requires the improvement of the numerical models describing the propagation between a radiating body and a microphone array. The commonly used free field transfer functions boil down to assume a full acoustic transparency of the radiating object, which, in some cases, may lead to misleading outcomes for their characterization. Among other approaches, equivalent sources methods (ESM) emerged as a convenient and powerful approach to simulate scattered sound fields. In this paper, an acoustic imaging algorithm, named Galerkin ESM, where equivalent sources are tailored to concomitantly match with microphone pressures and a Neumann boundary condition, is proposed. By means of a projected matrix inversion and backpropagation of the equivalent sources, Galerkin ESM aims at the direct synthesis of the pressure field around a diffracting body by making the most of an array measurement. This method is compared with two other existing imaging algorithms fueled by free field and computed transfer functions. The impact of the chosen transfer model is discussed, and Galerkin ESM performances are evaluated based on numerical and experimental test cases.

Short-time least squares spectral analysis of pass-by noise in water from a rigid inflatable boat.

This paper investigates least squares spectral analysis as a tool to analyze non-stationary signals from pass-by noise measurements in water. The spectral analysis involves successive least squares fitting of a finite Fourier series to approximate the observation in a piecewise manner. The least squares spectral analysis is used to search the signals for first- and second-order periodicity as well as the presence of fundamental periodicity. A first-order analysis reveals line components in the signals, whereas a second-order analysis reveals periodic amplitude modulations. Analysis with a higher-order finite Fourier series reveals harmonic structures in the signals. The main contribution of this paper is the model of a magnitude-squared cosine wave which can be used to analyze second-order periodicity. The developed short-time least squares spectral analysis is illustrated on noise radiated from a rigid inflatable boat in shallow water.

A Bayesian approach to eliminate correlated noise using an independent reference-Application to supersonic jet noise extraction.

A Bayesian method to remove correlated noise from multi-channel measurements is introduced. It is based on Bayesian factor analysis coupled with prior but uncertain knowledge of the correlation structure of the noise. This technique is well suited to denoise cross-spectral matrices measured in the frame of aeroacoustic experiments when background noise measurements are available, because it allows separating the engine noise contribution from the turbulent boundary layer and uniform noise components that are all sensed by in-flow microphones. In-flight data measured on flush-mounted microphones on an aircraft fuselage are denoised using this method. It is shown that it has a significant benefit for studying the broadband shock-associated noise generated by the engines in realistic flight conditions.

A Bayesian approach for the separation of the acoustic and the correlated aerodynamic wall pressure fluctuations.

When performing measurements with wall-installed microphone array, the turbulent boundary layer that develops over the measuring system can induce pressure fluctuations that are much greater than those of acoustic sources. It then becomes necessary to process the data to extract each component of the measured field. For this purpose, it is proposed in this paper to decompose the measured spectral matrix into the sum of matrices associated with the acoustic and aerodynamic contributions. This decomposition exploits the statistical properties of each pressure field. On the one hand, assuming that the acoustic contribution is highly correlated over the sensors, the rank of the corresponding cross-spectral matrix is limited to a finite number. On the other hand, the correlation structure of the aerodynamic noise matrix is constrained to resemble a Corcos-like model, with physical parameters estimated within the separation procedure. This separation problem is solved by a Bayesian inference approach, which takes into account the uncertainties on each component of the model. The performance of the method is first evaluated on wind tunnel measurements and then on a particularly noisy industrial measurement setup: microphones flush-mounted on the fuselage of a large aircraft.

A probabilistic approach for cross-spectral matrix denoising: Benchmarking with some recent methods.

Array measurements can be contaminated by strong noise, especially when dealing with microphones located near or in a flow. The denoising of these measurements is crucial to allow efficient data analysis or source imaging. In this paper, a denoising approach based on a Probabilistic Factor Analysis is proposed. It relies on a decomposition of the measured cross-spectral matrix (CSM) using the inherent correlation structure of the acoustical field and of the flow-induced noise. This method is compared with three existing approaches, aiming at denoising the CSM, without any reference or background noise measurements and without any information about the sources of interest. All these methods make the assumption that the noise is statistically uncorrelated over the microphones, and only one of them significantly impairs the off-diagonal terms of the CSM. The main features of each method are first reviewed, and the performances of the methods are then evaluated by way of numerical simulations along with measurements in a closed-section wind tunnel.

Mapping uncertainties involved in sound source reconstruction with a cross-spectral-matrix-based Gibbs sampler.

The reconstruction of sound sources by using inverse methods is known to be prone to estimation errors due to measurement noise, model mismatch, and poor conditioning of the inverse problem. This paper introduces a solution to map the estimation errors together with the reconstructed sound sources. From a Bayesian perspective, it initializes a Gibbs sampler with the Bayesian focusing method. The proposed Gibbs sampler is shown to converge within a few iterations, which makes it realistic for practical purposes. It also turns out to be very flexible in various scenarios. One peculiarity is the capability to directly operate on the cross-spectral matrix. Another one is to easily accommodate sparse priors. Eventually, it can also account for uncertainties in the microphone positions, which reinforces the regularization of the inverse problem.

Instantaneous Bayesian regularization applied to real-time near-field acoustic holography.

Real-time near-field acoustic holography (RT-NAH) is used to recover non-stationary sound sources using a planar microphone array. Direct propagation is described by the convolution of the wavenumber spectrum of the source under study with a known impulse response. The deconvolution operation is achieved by a singular value decomposition of the propagator and Tikhonov regularization is performed to stabilize the solution. The inverse problem has an innate ill-posed characteristic, and the regularization process is the key factor in obtaining acceptable results. The purpose of this paper is to present the instantaneous regularization process applied to RT-NAH method. Bayesian estimation of the regularization parameter is introduced from prior knowledge of the problem. The computation of the regularization parameter is updated for each block of constant time interval allowing one to take into account the fluctuating properties of the sound field. The superiority of Bayesian regularization, compared to state-of-the art methods, is observed numerically and experimentally for reconstruction of non-stationary sources. RT-NAH is also enhanced to allow the reconstruction of long signals. Updating the regularization parameter accordingly to the fluctuations of the SNR is revealed to be a necessary effort to reconstruct highly non-stationary sources.

Bayesian space-frequency separation of wide-band sound sources by a hierarchical approach.

This paper proposes an efficient solution to the separation of uncorrelated wide-band sound sources which overlap each other in both space and frequency domains. The space-frequency separation is solved in a hierarchical way by (1) expanding the sound sources onto a set of spatial basis functions whose coefficients become the unknowns of the problem (backpropagation step) and (2) blindly demixing the coefficients of the spatial basis into uncorrelated components relating to sources of distinct physical origins (separation step). The backpropagation and separation steps are both investigated from a Bayesian perspective. In particular, Markov Chain Monte Carlo sampling is advocated to obtain Bayesian estimates of the separated sources. Separation is guaranteed for sound sources having different power spectra and sufficiently smooth spatial modes with respect to frequency. The validity and efficiency of the proposed separation procedure are demonstrated on laboratory experiments.

Acoustical and mechanical characterization of poroelastic materials using a Bayesian approach.

A characterization method of poroelastic materials saturated by air is described. This inverse method enables the evaluation of all the parameters with a simple measurement in a standing wave tube. Moreover, a Bayesian approach is used to return probabilistic data such as the maximum a posteriori and the confidence interval of each parameter. To get these data, it is necessary to define prior probability distributions on the parameters characterizing the studied material. This last point is very important to regularize the inverse problem of identification. In a first step, the direct problem formulation is presented. Then, the inverse characterization is developed and applied to simulated and experimental data.

A Bayesian approach to sound source reconstruction: optimal basis, regularization, and focusing.

The reconstruction of acoustical sources from discrete field measurements is a difficult inverse problem that has been approached in different ways. Classical methods (beamforming, near-field acoustical holography, inverse boundary elements, wave superposition, equivalent sources, etc.) all consist--implicitly or explicitly--in interpolating the measurements onto some spatial functions whose propagation are known and in reconstructing the source field by retropropagation. This raises the fundamental question as whether, for a given source topology and array geometry, there exists an optimal interpolation basis which minimizes the reconstruction error. This paper provides a general answer to this question, by proceeding from a Bayesian formulation that is ideally suited to combining information of physical and probabilistic natures. The main findings are the followings: (1) The optimal basis functions are the M eigen-functions of a specific continuous-discrete propagation operator, with M being the number of microphones in the array. (2) The a priori inclusion of spatial information on the source field causes super-resolution according to a phenomenon coined "Bayesian focusing." (3) The approach is naturally endowed with an internal regularization mechanism and results in a robust regularization criterion with no more than one minimum. (4) It admits classical methods as particular cases.

Orthogonal-like fractional-octave-band filters.

This paper addresses the design of digital fractional-octave-band filters with energy conservation and perfect reconstruction--i.e., whose outputs to each fractional-octave-band correctly sum up to the original signal and whose partial energies at each output correctly sum up to the overall signal energy--a combination of properties that cannot be met by any current design despite its considerable importance in many applications. A solution is devised based on the introduction of complex basis functions that span the outputs of the fractional-octave bands and whose real and imaginary parts form two individually--but not mutually--orthogonal bases. This imposes a "partition-of-unity" condition on the design of the filter frequency gains such that they exactly sum up to one over the frequency axis. The practical implementation of the proposed solution uses the discrete Fourier transform, and a fast algorithm is implemented using the fast Fourier transform. The proposed filters are well suited to any application involving the post-processing of finite-energy signals. They closely match the international standard templates, except for a small departure at the bandedge frequencies which can be made arbitrarily small.

Analysis of muscular fatigue during cyclic dynamic movement.

We have investigated the electromyographic manifestations of fatigue on the Biceps Brachii in prolonged isometric and dynamic contractions by using two spectral analyses (Fourier transform (FT) and continuous wavelet transform (CWT)) as references and a novel approach cyclostationarity analysis. The isometric fatigue test shows no deviations from what is found in the literature (increase in energy (En) and decrease in mean frequency (MF)), while the dynamic fatigue test shows a slight increase in En and no change in MF for both spectral estimation techniques. In addition, the cyclostationarity increases with the fatigue and it could provide a new index of the fatigue during cyclostationary dynamic movements.