Non-stationary Sound Field Research Articles

Reconstruction of nonstationary sound field based on group sparsity time domain equivalent source method

To improve the accuracy and stability of reconstruction of nonstationary sound field, a novel time domain equivalent source method based on non-convex overlapping group sparsity (NOGS-TESM) is proposed. In the proposed NOGS-TESM, the time-varying pressure in the near field is first modeled by a set of equivalent sources positioned inside the vibrating structure. Then, the NOGS regularization is applied for solving the inverse problem, and a novel algorithm derived on the basis of Majorize-Minimization theory is utilized to obtain the optimal solution of equivalent sources strengths. Finally, the optimal equivalent sources strengths at all times are substituted into a time-domain convolution to calculate the time-varying pressure of reconstruction surface. The proposed NOGS-TESM enriches the prior information by considering the group sparsity of spatial pressure distribution, and enhances the accuracy of estimated equivalent sources strength via its alternative calculation of l0−norm. A vibrating plate to produce a nonstationary sound field, which is designed as a numerical simulation, and the time-varying pressures of measurement surface are input into the proposed NOGS-TESM. The quantitative and qualitative comparisons confirm that the proposed NOGS-TESM can greatly reconstruct the non-stationary sound field in time–space domains, and its reconstruction accuracy is higher than that of TESM with Tikhonov regularization (TR-TESM). Meanwhile, several parameters, for example, the noise and position of equivalent source surface, are discussed in detail, and the superiorities of the proposed NOGS-TESM in terms of stability and robustness are proved by comparing to TR-TESM. The predominance of the proposed NOGS-TESM is further validated by conducting one experiment on the vibrating steel plate.

Time-domain sound field reconstruction using a rigid spherical microphone array.

A time-domain approach for interior spherical near-field acoustic holography is proposed to achieve the low-delay reconstruction of time-domain sound fields using a rigid spherical microphone array. This reconstruction encompasses the incident pressure field, the incident radial particle velocity field, and the total pressure field, which includes scattering. The proposed approach derives time-domain radial propagators through the inverse Fourier transform of their frequency-domain counterparts. These propagators are then applied to the array measurements to obtain the time-domain spherical harmonic coefficients of the interior sound field. Given the fact that the time-domain radial propagators possess finite-time support and exhibit significant high-frequency attenuation characteristics, they can be efficiently implemented using finite impulse response (FIR) filters. The proposed approach processes the signal sample-by-sample through these FIR filters, avoiding a series of issues associated with time-frequency transformations in frequency-domain methods. As a result, the approach offers higher accuracy and lower latency in reconstructing non-stationary sound fields compared to its frequency-domain counterpart and thus holds greater potential for real-time applications. Additionally, owing to the scattering effect of the rigid sphere, the approach avoids the impact of spherical Bessel function nulls and does not require the measurement of particle velocities, which renders the measurements cost effective.

Reconstruction and analysis of a non-stationary sound field in a uniformly flow medium

A multistep time domain plane wave superposition method (M-TPWM) is expanded into a new environment that the medium is fluid, and an M-TPWM with flow medium (FM-TPWM) is proposed to reconstruct the non-stationary sound field in a uniformly flow medium. In the FM-TPWM, two situations that the flow directions of the medium are parallel to and perpendicular to the hologram plane are researched. The time-varying pressures on the hologram plane in these two situations are expressed as a superposition of time convolutions between the pressure time-wavenumber spectra of a virtual source plane and the corresponding new time domain propagation kernel derived in the flow medium. Subsequently, the discrete solving equations in the parallel and perpendicular situations at one time are established. The pressure time-wavenumber spectra at only this time can be solved by once solving inverse calculation. Then, these discrete solving equations in the flow medium at several times are combined for forming the larger solving matrix equations in the parallel and perpendicular situations, and by once multistep calculation, the pressure spectra at these times are acquired concurrently. Consequently, the time-varying pressures on the reconstruction plane in the flow medium are calculated by the corresponding acquired pressure time-wavenumber spectra. Numerical simulations with parallel and perpendicular situations are applied to examine the performance of the proposed FM-TPWM, in which two monopole sources in the flow medium are used to engender the non-stationary sound field. The simulation results prove that the proposed FM-TPWM can reconstruct the non-stationary sound field very well when the flow directions of the medium are parallel to and perpendicular to the hologram plane, respectively. Besides, several meaningful parameters are discussed in detail to prove the adaptability and robustness of the proposed FM-TPWM in the flow medium, and its reconstruction advantages are protruded in comparison to the real-time nearfield acoustic holography with the flow medium.

Separation of nonstationary sound fields based on the time-domain equivalent source method with single layer pressure-velocity measurements.

This paper proposes a sound field separation technique based on the time-domain equivalent source method with single layer pressure-velocity measurements to extract the nonstationary sound field radiated by the target source in a reverberant environment. This technique constructs a formulation that relates the pressures and particle velocities on a measurement surface to the strengths of time-domain equivalent sources arranged for modelling the outgoing and incoming waves. By solving the strengths of time-domain equivalent sources, the sounds coming from different sides of the measurement surface can be separated independently. In the proposed technique, the use of a time-domain equivalent source model allows the measurement surface to be arbitrarily shaped, thus providing the ability to analyze the arbitrarily shaped sources in a reverberant environment. Numerical simulations investigated the performance of the proposed technique when using different types of arrays, including planar, semi-cylindrical, and semi-spherical arrays, and an experiment with three loudspeakers located at two sides of the measurement surface was carried out to test the validity of the proposed technique. Both numerical and experimental results demonstrate that the proposed technique can remove the influence of disturbing sources in both time and space domains and separate out the target sound fields effectively.

Reconstruction of nonstationary sound fields based on a time domain angular spectrum method.

A time domain angular spectrum method is proposed to reconstruct nonstationary sound fields. In this method, the sound field is expressed as a superposition of a series of plane wave bases, and the plane wave basis is constructed by an impulse response function that relates the time domain angular spectrum to the field point pressure. The impulse response function consists of two parts, the propagating plane waves and the evanescent plane waves, and their physical interpretation is provided. By discretizing the time convolution between the plane wave strength and the impulse response function, the reconstruction can be carried out at each time step, thus providing the advantage of real-time reconstructing sound fields. Since the real-time reconstruction process is non-recursive, it can provide a stable reconstruction. In the reconstruction process, the Tikhonov regularization is introduced at each time step to obtain an appropriate estimation of the plane wave strength. Numerical simulations with an unsteady excitation plate and an experiment with an impacted plate were carried out to demonstrate the feasibility of the proposed method on reconstructing nonstationary sound fields. The effect of numerical parameters on the reconstruction accuracy was also investigated in the numerical simulations.

Real-time separation of non-stationary sound fields on spheres.

The sound field separation methods can separate the target field from the interfering noises, facilitating the study of the acoustic characteristics of the target source, which is placed in a noisy environment. However, most of the existing sound field separation methods are derived in the frequency-domain, thus they are best suited for separating stationary sound fields. In this paper, a time-domain sound field separation method is developed that can separate the non-stationary sound field generated by the target source over a sphere in real-time. A spherical array sets up a boundary between the target source and the interfering sources, such that the outgoing field on the array is only generated by the target source. The proposed method decomposes the pressure and the radial particle velocity measured by the array into spherical harmonic coefficients, and recovers the target outgoing field based on the time-domain relationship between the decomposition coefficients and the theoretically derived spatial filter responses. Simulations show the proposed method can separate non-stationary sound fields both in free field and room environments, and over a longer duration with small errors. The proposed method could serve as a foundation for developing future time-domain spatial sound field manipulation algorithms.

Recovery of non-stationary free field based on pressure and acceleration measurements in a noisy environment

In a noisy environment, the interested sound field is interfused with the non-stationary incoming field from the back side of the measurement plane and the scattered field caused by the incoming wave falling on the surface of the target source. In order to remove the non-stationary incoming and scattered fields simultaneously, a recovery method of the non-stationary free field with the pressure and particle acceleration measurements is proposed. First, the mixed time-evolving pressure and particle acceleration are firstly measured on one measurement plane, where the particle acceleration is obtained by the finite difference approximation with the aid of an auxiliary measurement plane; Then, two physical relations are employed to deduce a forward complete recovery formulation of the target source in the noisy environment. One relation contains two impulse response functions relating the time-wavenumber pressure spectrum to particle acceleration and pressure spectra, respectively, and the other is the surface reflection coefficient of the target source relating the scattered field to the incoming field. Finally, the mixed pressure and particle acceleration are substituted into the recovery formulation, and the time-evolving pressure radiated by the target source in free-field is recovered. Thereby, the proposed method possesses a significant feature of real-time recovery of the non-stationary free field. A circular piston fixed on an infinite rigid baffle and two monopole sources are designed in a numerical simulation to test the performance of the proposed method. The simulation results attest that the proposed method not only can recover the time-evolving pressure radiated by the target source in free-field at any space points, but also recover the space distribution of the non-stationary sound field of the target source in free-field at different time instants effectively. An experiment with two speakers embedded in a planar steel plate and a speaker is further employed to verify the validity of the proposed method.

A multistep time-domain plane wave superposition method for stabilizing the reconstruction of the non-stationary sound field

Abstract A multistep time domain plane wave superposition method (M-TDPWSM) is proposed to stabilize the reconstruction process of a non-stationary sound field when the measured time-evolving pressure on the hologram plane is used to solve the time-wavenumber pressure spectra on the virtual source plane. Unlike the conventional TDPWSM that performs each solving inverse calculation for only one time step in reconstructing the time-evolving pressure, the proposed M-TDPWSM combines the solving equations at several time steps to form a large matrix equation for once solving inverse calculation. The multistep calculation in the proposed M-TDPWSM not only reduces the total number of the solving inverse calculation for decreasing the accumulation errors, but also changes the element distribution of the inverse matrix to improve the filtering effect. One numerical simulation with an exciting plate is designed to examine the abilities of the proposed M-TDPWSM and the conventional TDPWSM. Several parameters including the position of the virtual source plane, the sampling number in the wavenumber domain, the noise and the number k of the proposed M-TDPWSM are investigated in the simulation. The simulation results attest that the reconstructed time-evolving pressure by the proposed M-TDPWSM at different parameter settings matches well with the theoretical one, but the conventional TDPWSM is sensitive to these parameters and sometimes its solutions are emanative. The proposed M-TDPWSM can suppress the numerical instability of reconstructing the non-stationary sound field and provide a stable solution. An experiment with an impacted planar plate was carried out in a semi anechoic room to further verify the superiority of the proposed M-TDPWSM in improving the stability of the solving inverse process.

Recovery of non-stationary free field with double-layer pressure measurements

Abstract To remove the non-stationary incoming field from the back side of the measurement plane and the non-stationary scattered field caused by the incoming wave falling on the surface of the target source, a recovery method of the non-stationary free field with double-layer pressure measurements is proposed. In this method, the time-evolving mixed pressures on double-layer measurement planes are first measured and transformed into the time-wavenumber domain. Then, with the aid of these time-wavenumber pressure spectrums and the impulse response function, the non-stationary incoming and outgoing fields are separated by the conventional non-stationary sound field separation method. Subsequently, the time-wavenumber spectrum of the surface reflection coefficient of the target source is employed to establish the relationship between the non-stationary scattered and incoming fields, based on which the non-stationary scattered field is calculated. Finally, the non-stationary scattered field is removed from the outgoing field and the non-stationary sound field that would be radiated by the target source in free field is recovered. Numerical simulation with an exciting planar steel plate as the target source and a monopole source as the disturbing source is used to examine the feasibility of the proposed method. The recovered results demonstrate that the proposed method not only removes the influence of the incoming field in the time and space domains, but also clears the non-stationary scattered field, thus providing more accurate recovered results than the non-stationary sound field separation method. The experiment further proves the validity of the proposed method.

Application of time-domain near-field acoustical holography to the high-order ambisonics presentation of moving sound sources

High-order ambisonics (HOA) is a sound field reproduction method based on results from near-field acoustical holography (NAH). As such, it is formulated in the frequency domain; this assumes a stationary acoustic field. This presentation applies a recent time-domain formulation of NAH by Attendu and Ross. The approach is based on the linear deconvolution along the spatial coordinates and samples the Green's function in the time and spatial domains, as opposed to the Fourier domain. Our contribution is an application of these ideas to overcome the problems inherent in the HOA reproduction of non-stationary sound fields, in particular those of moving sound sources. In practice, the rendering of moving sound sources is approximated by assuming they are stationary within short temporal windows. The approximation, while useful for slow-moving sources, cannot re-create the Doppler shift and results in discontinuities along the temporal structure of the sound field. On the other hand, the proposed method does not impose any constraints on the motion of the sound sources. Unlike conventional HOA, the proposal cannot be formulated as a simple filter bank; however, numerical experiments show that fast moving sources exhibit the expected Doppler shift.

Real-time nearfield acoustic holography in a uniformly moving medium

Real-time nearfield acoustic holography (RT-NAH) is an effective tool to identify nonstationary sound sources and predict the time-dependent sound field via a temporal convolution between the time-dependent wavenumber spectrum on the hologram plane and an impulse response. However, the conventional RT-NAH procedures are developed for sound sources situated in a static medium. As for sound sources located in a moving medium, the conventional RT-NAH procedures cannot be applied directly due to the fact that the impulse response will be changed by flow effects. In this paper, two analytical impulse responses in a uniformly moving medium, corresponding to two cases that the flow direction is parallel to and perpendicular to the hologram plane, are derived first with consideration of flow effects, and then RT-NAH is extended to realize forward and backward propagation of time-dependent signals in the moving medium. Numerical simulations are conducted to check the performances of the proposed method. The results show that the proposed method not only can be used to predict nonstationary sound fields but also can be utilized to identify nonstationary sources in a moving medium for both the parallel and perpendicular cases.

Time domain nearfield acoustical holography without wrap-around error and spectral leakage for forward propagation.

Nearfield acoustical holography can be used to visualize time domain evolution of non-stationary sound fields. It requires sampling of the pressure field with a microphone array to calculate the sound field on other parallel surfaces, either toward the source or farther away from it. This paper focuses on forward propagation of fields sampled with a planar array. Several formulations exist to perform such calculation; however, despite being mathematically well-posed, they generate significant errors, especially for larger propagation distances. These errors are mainly due to wrap-around and spectral leakage. It is shown in this paper that generation of leakage can be avoided by sampling the analytical Green's function in the time and space domains. Such formulation allows the application of linear convolution, which completely suppresses wrap-around errors. The results obtained from the proposed formulation are compared to the analytical solution of a baffled piston with transient displacement. The average discrepancy is below 3.5% for propagation distances of 5 to 55 cm. In comparison, the relative error obtained with other propagators varies from 6% to 73%, depending on the propagation distance and method used. Finally, the proposed formulation is applied to experimental data obtained from an impacted plate.

Separation of non-stationary sound fields with single layer pressure-velocity measurements.

This paper examines the feasibility of extracting the non-stationary sound field generated by a target source in the presence of disturbing source from single layer pressure-velocity measurements. Unlike the method described in a previous paper [Bi, Geng, and Zhang, J. Acoust. Soc. Am. 135(6), 3474-3482 (2014)], the proposed method allows measurements of pressure and particle velocity signals on a single plane instead of pressure signals on two planes, and the time-dependent pressure generated by the target source is extracted by a simple superposition of the measured pressure and the convolution between the measured particle velocity and the corresponding impulse response function. Because the particle velocity here is measured directly, the error caused by the finite difference approximation can be avoided, which makes it possible to perform the separation better than the previous method. In this paper, a Microflown pressure-velocity probe is used to perform the experimental measurements, and the calibration procedure of the probe in the time domain is given. The experimental results demonstrate that the proposed method is effective in extracting the desired non-stationary sound field generated by the target source from the mixed one in both time and space domains, and it obtains more accurate results than the previous method.

Real-time separation of non-stationary sound fields with pressure and particle acceleration measurements.

To extract the desired non-stationary sound field generated by a target source in the presence of disturbing sources, a real-time sound field separation method with pressure and particle acceleration measurements is proposed. In this method, the pressure and particle acceleration signals at a time instant are first measured on one measurement plane, where the particle acceleration is obtained by the finite difference approximation with the aid of an auxiliary measurement plane; then, the desired pressure signal generated by the target source at the same time instant can be extracted in a timely manner, by a simple superposition of the measured pressure and the convolution between the measured particle acceleration and the derived impulse response function. Thereby, the proposed method possesses a significant feature of real-time separation of non-stationary sound fields, which provides the potential to in situ analyze the radiation characteristics of a non-stationary source. The proposed method was examined through numerical simulation and experiment. Results demonstrated that the proposed method can not only extract the desired time-evolving pressure signal generated by the target source at any space point, but can also obtain the desired spatial distribution of the pressure field generated by the target source at any time instant.

Near-Field Acoustic Holography of Cyclostationary Sound Field

The radiated sound field of rotating machinery or reciprocating machinery has a significant periodically time-variant nature. This is a kind of non-stationary sound field and called cyclostationary sound field. In the conventional planar near-field acoustic holography(PNAH), this kind of sound field is treated as stationary field, so the information relating to the change of frequency with time will be loss inevitably. In this article, the cyclic spectral density(CSD) instead of the complex sound pressure was adopted as reconstructing physical quantity in the PNAH, and the cyclostationary PNAH(CPNAH) technique was proposed. Meanwhile, focusing on the calculation complex of CSD and the accuracy of the cyclic nature extracted, the gathering slice method of CSD was proposed by referring time aliasing methods on time series. The experiment results illustrate that the cyclic nature of cyclostationary sound field may be extracted directly and the location of the source determined exactly as well.

Reconstruction of nonstationary sound fields based on the time domain plane wave superposition method

A time-domain plane wave superposition method is proposed to reconstruct nonstationary sound fields. In this method, the sound field is expressed as a superposition of time convolutions between the estimated time-wavenumber spectrum of the sound pressure on a virtual source plane and the time-domain propagation kernel at each wavenumber. By discretizing the time convolutions directly, the reconstruction can be carried out iteratively in the time domain, thus providing the advantage of continuously reconstructing time-dependent pressure signals. In the reconstruction process, the Tikhonov regularization is introduced at each time step to obtain a relevant estimate of the time-wavenumber spectrum on the virtual source plane. Because the double infinite integral of the two-dimensional spatial Fourier transform is discretized directly in the wavenumber domain in the proposed method, it does not need to perform the two-dimensional spatial fast Fourier transform that is generally used in time domain holography and real-time near-field acoustic holography, and therefore it avoids some errors associated with the two-dimensional spatial fast Fourier transform in theory and makes possible to use an irregular microphone array. The feasibility of the proposed method is demonstrated by numerical simulations and an experiment with two speakers.

Near-field acoustic holography of cyclostationary sound fields

The radiated sound field of rotating machinery or reciprocating machinery has a significant periodically time-variant nature. This is a kind of non-stationary sound field and called cyclostationary sound field. In the conventional planar near-field acoustic holography(PNAH), this kind of sound field is treated as stationary field, so the information relating to the change of frequency with time will be loss inevitably. In this article, the cyclic spectral density(CSD) instead of the complex sound pressure was adopted as reconstructing physical quantity in the PNAH, and the cyclostationary PNAH(CPNAH) technique was proposed. Meanwhile, focusing on the calculation complex of CSD and the accuracy of the cyclic nature extracted, the gathering slice method of CSD was proposed by referring time aliasing methods on time series. The experiment results illustrate that the cyclic nature of cyclostationary sound field may be extracted directly and the location of the source determined exactly as well.

Separation of nonstationary sound fields in the time-wavenumber domain

A number of sound field separation techniques have been proposed for different purposes. However, these techniques just consider the separation of sound fields in the space domain and are restricted to stationary sound fields. When the sound fields are nonstationary, it is also necessary to perform the separation in the time domain. Therefore, on the basis of the propagation principle of sound pressure in the time-wavenumber domain, a nonstationary sound field separation technique with two closely spaced parallel measurement surfaces is proposed. It can separate the nonstationary signals generated by the primary sources in both time and space domains when the disturbing sources exist on the other side of the measurement plane. The signals in time and space domains are separated by using the spatial Fourier transform method and the time domain deconvolution method. A simulation involving two monopoles driven by nonstationary signals demonstrates that the method proposed can remove the influence of disturbing sources in both time and space domains. The feasibility of this method is also demonstrated by an experiment with two loudspeakers located on two sides of measurement planes. Additionally, to comment more objectively on the separation results, some indicators are computed in both the simulation and experiment.