Plane Wave Decomposition Research Articles

Cosmic microwave background in a non-Archimedean world

The universe begins its expansion from dimensions below the Planck scale, where the uncertainty principle reigns. If at these dimensions the Archimedean geometry is useless, the p-adic non-Archimedean world creates an environment conducive to phenomenological modeling. The standard description of the cosmological density contrast is made by the Fourier decomposition of plane waves in the field of p-adic numbers Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document}. The coupling of the primordial perturbations is done by coupling the phases. Since the phases are the stars of this article, we represent them in the hue-saturation-brightness color space both in R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document} and in Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document}. If the primordial perturbations interact weakly with each other, with their growth they become important enough from an energetic point of view to gravitationally interact with each other. We identify the coupling using the phase gradient operator, and extract the coupled points from the spectrum. We use pseudoentropy for the quantitative analysis of the information generated by the phases.

Multizone sound field reproduction based on equivalent source decomposition

Multizone sound field reproduction attempts to recreate a desired sound field on the bright zone and reduce sound field energy on the dark zone. Conventional pressure-matching based multizone reproduction method is only able to control sound pressures at limited number of measurement points on the bright zones which are equal to target sound field recording points number. The reproduction performance is strongly affected by the number of recording points. In this paper, a reproduction method based on the sparse plane wave decomposition method is proposed to improve the reproduction performance when the target sound field is generated by one or a small number of sources. The recorded target sound field is decomposed into a group of plane waves. Loudspeaker weights are calculated as a summation precalculated corresponding plane waves weights. A simulation is conducted in a room to verify the performance of the proposed method. Compared with the conventional method, the proposed method achieves better sound field reproduction accuracy on the bright zone and similar acoustic contrast performance.

Numerical Modeling and Geometry Enhancement of a Reactive Silencer

Internal combustion engines and blowers frequently utilize silencers to reduce exhaust noise. In the current paper, the transmission loss of reactive silencers is predicted using the plane wave decomposition method and a three-dimensional (3-D) time-domain computational fluid dynamics (CFD) approach. A mass-flow-inlet boundary condition is first used to perform a steady flow computation, which serves as an initial condition for the two subsequent unsteady flow computations. At the model's inlet, an impulse (acoustic excitation) is placed over the constant mass flow to perform the first unstable flow computation. Once the impulse has fully propagated into the silencer, the non-reflecting boundary condition (NRBC) is then added. For the scenario without acoustic excitation at the inlet, a second unsteady flow computation is performed. During the two transient computations, the time histories of the pressure and velocity at the upstream measuring points as well as the history of the pressures at the downstream measuring point are recorded. The related acoustic quantities show variations between the two unsteady flow computational findings. As a result, the transmitted sound pressure signal is just the sound pressure downstream, while the incident sound pressure signal is obtained by utilizing plane wave decomposition upstream. The transmission loss (TL) of the silencer is then calculated after the Fast Fourier Transform (FFT) converts the two sound pressure signals from the time domain to the frequency domain. The numerical calculations and the reported data are in good agreement for the published results, in addition to geometry enhancement by increasing number of holes in the cross section for muffler.

Green’s function for thick laminated composite plates with coupled stretching-bending and transverse shear deformation

In this paper, the Green’s function of a thick laminated composite plate is derived for the first time in the literature. The plate can be unsymmetric with respect to the mid-plane such that the coupling of in-plane stretching and out-of-plane bending deformations occurs. Also, the plate is thick enough that the transverse shear deformation cannot be ignored. Through the plane wave decomposition method, the partial differential governing equations based upon the first order shear deformation plate theory can be reorganized into the ordinary differential equations in terms of the transformed variable. With the governing equations reorganized into matrix form, the Green’s functions are expressed in terms of matrix exponential. Through eigen-decomposition, the matrix exponential including its related integrals can be evaluated explicitly. The explicit solutions of Green’s functions in transformed domain are obtained accordingly. In order to improve the accuracy and efficiency of numerical integration for the required transform integrals, a special quadrature rule which can eliminate the integral singularity is suggested. To have a fair verification and to extend the applicability of newly derived Green’s functions, the associated boundary element method is also introduced in this study.

Harvey–Shack theory for a converging–diverging Gaussian beam

The scattering characteristics of random rough surfaces illuminated with a 3D converging–diverging Gaussian beam are investigated by applying the conventional Harvey–Shack theory in conjunction with 2D plane-wave decomposition. The Gaussian beam is assumed to have an arbitrary angle of incidence and to be linearly s-polarized. Using data obtained from laser BRDF measurements on isotropic random rough surfaces with low surface roughness, we demonstrate that the Gaussian beam Harvey–Shack theory is in better accordance with the experimental data than the conventional Harvey–Shack theory. The two models become identical for a large beam waist radii but are significantly different for smaller ones.

Bayesian selection of plane-wave decomposition models.

Plane-wave decompositions, whereby a measured sound field is described as a superposition of plane waves, are central to many applications in acoustics and audio engineering. This letter applies a Bayesian probabilistic inference framework to the plane wave decomposition problem and examines the Deviance Information Criterion (DIC) for selecting the optimum number of waves in the decomposition. The framework learns the model directly from the data and, as such, adapts to the wavefield under study. The DIC is applied to data measured in two reverberant sound fields (highly-reverberant and lightly-damped) to determine the simplest models providing the preferred fit to the data.

On the Connection Between the Fueter–Sce–Qian Theorem and the Generalized CK-Extension

The Fueter–Sce–Qian theorem provides a way of inducing axial monogenic functions in $$\mathbb {R}^{m+1}$$ from holomorphic intrinsic functions of one complex variable. This result was initially proved by Fueter and Sce for the cases where the dimension m is odd using pointwise differentiation, while the extension to the cases where m is even was proved by Qian using the corresponding Fourier multipliers. In this paper, we provide an alternative description of the Fueter–Sce–Qian theorem in terms of the generalized CK-extension. The latter characterizes axial null solutions of the Cauchy–Riemann operator in $$\mathbb {R}^{m+1}$$ in terms of their restrictions to the real line. This leads to a one-to-one correspondence between the space of axially monogenic functions in $$\mathbb {R}^{m+1}$$ and the space of analytic functions of one real variable. We provide explicit expressions for the Fueter–Sce–Qian map in terms of the generalized CK-extension for both cases, m even and m odd. These expressions allow for a plane wave decomposition of the Fueter–Sce–Qian map or, more in particular, a factorization of this mapping in terms of the dual Radon transform. In turn, this decomposition provides a new possibility for extending the Coherent State Transform (CST) to Clifford Analysis. In particular, we construct an axial CST defined through the Fueter–Sce–Qian mapping, and show how it is related to the axial and slice CST’s already studied in the literature.

Ambisonics decoder design based on sparse plane wave decomposition

Sound field reproduction by loudspeaker array is an important way to create acoustical virtual reality. High order ambisonics (HOA) is a highly applicable method to reproduce a desired sound field with a loudspeaker array. For a target sound field that is represented by a group of harmonic coefficients, the process of calculating loudspeaker weights for reproduction with these harmonic coefficients is called decoding. Mode-matching based decoding is usually applied in HOA reproduction and good performance is achieved in the loudspeaker array center (the ‘sweet spot’). However, the accuracy and sweet spot size are strongly affected by the recording order of the target sound field. To improve the reproduction performance, a new ambisonic decoding method is proposed based on the sparse plane wave decomposition in the case that the target sound field is steered by a small number of acoustic sources in this paper. The recorded target sound field is decomposed into a group of plane waves in the harmonic domain. Loudspeaker weights are calculated as a summation of the precalculated corresponding plane waves weights. Simulations are conducted in the free field and reverberation room to verify that the proposed method outperforms the mode-matching decoding. Experimental validation is conducted in a real listening room with a circular loudspeaker array. It is concluded that the reproduction error by the proposed method is superior to that by the sectorial mode matching for the investigated 2.5D reproduction case in the reverberant environment after 500 Hz and in terms of the MAC values, the proposed method ensures the spatial correspondence over frequency of the desired sound field and the reproduced sound field.

Manifold learning for global active noise control: Sensor interpolation

To achieve global control, unmeasured frequency responses (FRs) of a control source and preselected control points are required. In this paper, sensor interpolation (SI) approaches based on plane wave decomposition (PWD), kernel ridge regression, deep neural network (DNN) are presented. A broadened control region encompassing a large number of measured and fictitious control points can be obtained using the SI techniques. PWD is a two-stage procedure that begins with trimming down plane wave components, followed by an extraction of the associated complex amplitudes. In kernel ridge regression, a reproducing kernel in the Hilbert space (RKHS) is utilized to interpolate continuous relative transfer functions. The DNN-based SI approach is inspired by the kernel method. Convolutional neural networks and the like are exploited to interpolate the FRs at the fictitious points. Simulations are conducted for a thirty-microphone uniform linear array. From the results, an excellent fit between the FRs at the fictitious points and the ground truth generated by the image source method can be obtained below the spatial aliasing frequency. The efficacy of the interpolated FRs in relation to global control is demonstrated via an example of feedforward active noise control using a linear loudspeaker array.

Feedforward active noise global control using a linearly constrained beamforming approach

Global control of noise in a three-dimensional space is generally difficult, given a limited number of discrete sensors. This paper presents a feedforward multichannel active noise control (ANC) approach to overcome this difficulty. In light of the model-matching principle, an underdetermined multichannel inverse filtering (UMIF) system is formulated. Infinite number of solutions exist to yield zero residual noise at the error microphone positions. By incorporating the UMIF system as a design constraint, a multichannel feedforward controller is formulated using the linearly constrained minimum variance (LCMV) approach in which a cost function is introduced to minimize the residual noise energy at a large number of preselected fictitious control points. To implement the ANC system in reverberant environments, a novel sensor interpolation technique based on plane wave decomposition is developed to find the frequency responses between secondary loudspeakers and the fictitious control points. Only a limited number of room frequency response measurements are required in the use of these techniques. Simulations and experiments are undertaken using a uniform linear loudspeaker array to validate the proposed ANC system. The results have demonstrated that the proposed approach yielded substantial noise reduction in a widened and connected control area.

Acoustic modal analysis of room responses from the perspective of state-space balanced realization with application to field interpolation.

Despite its importance in structural dynamics and vibration, modal analysis is rarely performed in acoustics due to the high modal density of sound fields. A novel acoustic modal analysis (AMA) approach is proposed in this paper for enclosed acoustic fields, such as a room, from the perspective of state-space formulation of control systems. A single-input-multiple-output (SIMO) state-space model is established in light of the balanced realization (BR), given impulse response measurements. The BR model is then converted to a modal form such that the modal parameters, including natural frequencies, damping ratios, and mode shapes, can be estimated. To reconstruct mode shapes, plane wave decomposition (PWD) and compressive sensing (CS) techniques are exploited to solve the underdetermined problem for a spatially sparse representation of mode shapes under the Schroeder frequency. As a result, a model of a continuous system can be "interpolated" for any arbitrary source-receiver positions on the basis of the estimated mode shapes. With the identified modal parameters, the low-frequency and early reflection part of room impulse responses (RIRs) can be synthesized for arbitrary source-sensor pairs. The proposed AMA acoustic field interpolation is validated by extensive simulations and experiments.

Measuring angle-dependent sound absorption in ordinary rooms

The effective use of sound absorbing materials in room acoustical design requires the use of angle-dependent coefficients, as opposed to random incidence coefficients, to properly account for the dissipation of sound energy at the room's boundary. In this study, a method is proposed for measuring the angle-dependent surface impedance and absorption coefficient of a boundary material in situ, in an ordinary room. The proposed method relies on the reconstruction of the pressure and normal particle velocity at the boundary using a plane-wave decomposition of the sound field measured with an array of microphones. In sufficiently reverberant environments, where sound strikes the material from nearly all directions in space, the angle-dependent properties of the boundary can be obtained simultaneously for all angles of incidence, from a single source position. In addition, the proposed methodology enables us to identify the specific directions of incidence on the material and makes it possible to estimate its operational acoustic performance. The validity of the method is examined experimentally in a reverberation chamber and in a classroom, using a robotic arm to scan the sound field in the vicinity of an absorbing boundary. Consistent absorption data are measured for various source positions and room configurations.

On the Radon transform and the Dirac delta distribution in superspace

In this paper, we obtain a plane wave decomposition for the delta distribution in superspace, provided that the superdimension is not odd and negative. This decomposition allows for explicit inversion formulas for the super Radon transform in these cases. Moreover, we prove a more general Radon inversion formula valid for all possible integer values of the superdimension. The proof of this result comes along with the study of fractional powers of the super Laplacian, their fundamental solutions, and the plane wave decompositions of super Riesz kernels.

Transmission–propagation–diffraction operator theory for an arbitrary acoustic two-block model

By applying the transmission–propagation–diffraction​ operator theory (TPDOT; in earlier publications abbreviated as TPOT) we have derived an analytical representation of the wavefield in two 3D inhomogeneous acoustic half-spaces separated by a piecewise-smooth interface. We scrutinize the conventional approach and reformulate the mechanical particle motion forward problem in terms of a novel wave amplitude form of counter-propagating wave events. The new wave motion statement is composed of two systems of integral equations describing propagation and transmission of the seismic wavefields. The first system represents the propagation–diffraction operator equation. We express the propagation–diffraction operator as the feasible Kirchhoff-type surface integral with a feasible fundamental solution in the kernel. The feasible fundamental solution generalizes the conventional free space Green’s function for an arbitrary domain case. The second system represents the transmission (refraction/reflection) operator equation. This convolutional interface equation generalizes the Sommerfeld–Weyl spatial-frequency plane wave decomposition in an infinitesimally thin interface layer. We transform the wave amplitude formulation by Neumann iteration into infinite series of counter-propagating multiply transmitted (reflected/refracted) wave modes, representing an analytical solution to the problem. Since the wave motion statement of the forward problem enables analysis of separate wave modes of the total wavefield, it has an advantage over the particle motion statement. The suggested wavefield representation therefore solves both forward and inverse wave motion problems. Calculation of separate wave modes of TPDOT can be done by the software package of the tip-wave superposition method (TWSM) introduced in our earlier papers (in co-authorship). The proposed TPDOT provides the theoretical foundations for TWSM.

Compensating first reflections in non-anechoic head-related transfer function measurements

Personalized Head-Related Transfer Functions (HRTFs) are needed as part of the binaural sound individualization process in order to provide a high-quality immersive experience for a specific user. Signal processing methods for performing HRTF measurements in non-anechoic conditions are of high interest to avoid the complex and inconvenient access to anechoic facilities. Non-anechoic HRTF measurements capture the effect of room reflections, which should be correctly identified and eliminated to obtain HRTFs estimates comparable to ones acquired in an anechoic setup. This paper proposes a sub-band frequency-dependent processing method for reflection suppression in non-anechoic HRTF signals. Array processing techniques based on Plane Wave Decomposition (PWD) are adopted as an essential part of the solution for low frequency ranges, whereas the higher frequencies are easily handled by means of time-crop windowing methods. The formulation of the model, extraction of parameters and evaluation of the method are described in detail. In addition, a validation case study is presented showing the suppression of reflections from an HRTF measured in a real system. The results confirm that the method allows to obtain processed HRTFs comparable to those acquired in anechoic conditions.

Diffraction Separation With Plane-Wave Destruction Filer for the Lunar Penetrating Radar Sensor Data

China transported the lunar rovers to the lunar surface via the Chang’E rockets for lunar exploration. An important detection tool in the lunar rover is the lunar penetrating radar (LPR). LPR can obtain the lunar surface structure information by transmitting and receiving high-frequency electromagnetic waves. There are many types of waves received by LPR sensors, among which diffracted waves are often ignored. The diffractions in the LPR data contain a lot of underground structure information, especially for faults and small targets with high resolution. A major technical problem in effectively using the information is how to separate diffractions from different types of wave field information. This paper focuses on the diffraction issue and employs the plane wave decomposition filter method to separate the diffractions in LPR data sets. First, the records are converted into a dip field, and then the non-diffraction waves in the dip field is filtered to achieve the purpose of separating the diffractions. Three numerical models will be used to verify the effectiveness of the diffraction separation method.

Generalized momentum conservation and Fedorov-Imbert linear shift of acoustic vortex beams at a metasurface

We study the transmission and reflection of a paraxial acoustic vortex beam (AVB) at a metasurface with a prescribed phase gradient. By using a plane-wave decomposition method, we obtain a closed-form expression for the transverse shift of the beam's gravity center. The transverse shift is proportional to the topological charge of the AVB and derives from momentum conservation and the reflection/transmission coefficient modulation. It corresponds to the acoustic Fedorov-Imbert linear shift and represents the orbital Hall effect for spinless longitudinal sound. We develop a generalized momentum conservation relation to understand the phenomena and find that the metasurface induces both linear momentum and angular momentum, which have different dependences on the incident angle. In particular, this mechanism can enable the Fedorov-Imbert linear shift even for normal incidence and total reflection. The results provide deeper insights into the physics underlying the interaction of AVBs with metasurfaces and may help the development of orbital angular momentum based acoustic devices.

On hybrid convolution quadrature approaches for modeling time‐domain wave problems with broadband frequency content

AbstractWe propose two hybrid convolution quadrature based discretizations of the wave equation on interior domains with broadband Neumann boundary data or source terms. The convolution quadrature method transforms the time‐domain wave problem into a series of Helmholtz problems with complex‐valued wavenumbers, in which the boundary data and solutions are connected to those of the original problem through the ‐transform. The hybrid method terminology refers specifically to the use of different approximations of these Helmholtz problems, depending on the frequency. For lower frequencies, we employ the boundary element method, while for more oscillatory problems, we develop two alternative high frequency approximations based on plane wave decompositions of the acoustic field on the boundary. In the first approach, we apply dynamical energy analysis to numerically approximate the plane wave amplitudes. The phases will then be reconstructed using a novel approach based on matching the boundary element solution to the plane wave ansatz in the frequency region where we switch between the low and high frequency methods. The second high frequency method is based on applying the Neumann‐to‐Dirichlet map for plane waves to the given boundary data. Finally, we investigate the effectiveness of both hybrid approaches across a range of numerical experiments.

Optimization of secondary source configuration in enclosure using plane wave decomposition

The optimization of secondary source configuration for an active noise control (ANC) system in its enclosed space generally focuses on noise reduction requirements at discrete points only. This may lead to the poor noise reduction performance in the whole spatial region, and it is necessary to know the information on error sensor positions in advance. To address this problem, a cost function for spatial-region-oriented noise reduction is proposed. The plane wave decomposition of the enclosed sound field is used to obtain the primary field plane waves and the unit secondary field plane wave of each candidate secondary source as the prior knowledge for configuration optimization, so as to formulate a wave-domain ANC cost function. The optimization method adopts the simulated annealing search. Taking a rigid-walled rectangular cavity as an example, the optimization method is firstly compared with two space-domain methods by using analytic values of the wave-domain prior knowledge. The comparison results show that the better reduction of spatial acoustic potential energy can be achieved independent of the error sensor configuration information. Then the estimated values of the wave-domain prior knowledge through measuring randomly distributed microphones are used to optimize the configuration of the ANC system. The optimization results suggest that the noise reduction of spatial acoustic potential energy of the optimized configuration can be better than that of the space-domain method, but the microphone positions have a great influence on the noise reduction performance.

Spherical harmonic decomposition of a sound field based on observations along the equator of a rigid spherical scatterer.

We present a method for computing a spherical harmonic representation of a sound field based on observations of the sound pressure along the equator of a rigid spherical scatterer. Our proposed solution assumes that the captured sound field is height invariant so that it can be represented by a two-dimensional (2D) plane wave decomposition (PWD). The 2D PWD is embedded in a three-dimensional representation of the sound field, which allows for perfectly undoing the effect of the spherical scattering object. If the assumption of height invariance is fulfilled, then the proposed solution is at least as accurate as a conventional spherical microphone array of the same spherical harmonic order, which requires a multiple of the number of sensors. Our targeted application is binaural rendering of the captured sound field. We demonstrate by analyzing the binaural output signals that violations of the assumptions that the solution is based on-particularly height invariance and consequently also horizontal propagation-lead to errors of moderate magnitude.