Harmonic Decomposition Research Articles

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.

Complexity reduction in the 3D Kuramoto model

The dynamics of large systems of coupled oscillators is a subject of increasing importance with prominent applications in several areas such as physics and biology. The Kuramoto model, where a set of oscillators move around a circle representing their phases, is a paradigm in this field, exhibiting a continuous transition between disordered and synchronous motion. Reinterpreting the oscillators as rotating unit vectors, the model was extended to allow vectors to move on the surface of D-dimensional spheres, with $D=2$ corresponding to the original model. It was shown that the transition to synchronous dynamics was discontinuous for odd D, raising a lot of interest. Inspired by results in 2D, Ott et al proposed an ansatz for density function describing the oscillators and derived equations for the ansatz parameters, effectively reducing the dimensionality of the system. Here we take a different approach for the 3D system and construct an ansatz based on spherical harmonics decomposition of the distribution function. Our result differs significantly from that proposed in Ott's work and leads to similar but simpler equations determining the dynamics of the order parameter. We derive the phase diagram of equilibrium solutions for several distributions of natural frequencies and find excellent agreement with simulations. We also compare the dynamics of the order parameter with numerical simulations and with the previously derived equations, finding good agreement in all cases. We believe our approach can be generalized to higher dimensions and help to achieve complexity reduction in other systems of equations.

Search for anisotropic gravitational-wave backgrounds using data from Advanced LIGO and Advanced Virgo’s first three observing runs

We report results from searches for anisotropic stochastic gravitational-wave backgrounds using data from the first three observing runs of the Advanced LIGO and Advanced Virgo detectors. For the first time, we include Virgo data in our analysis and run our search with a new efficient pipeline called {\tt PyStoch} on data folded over one sidereal day. We use gravitational-wave radiometry (broadband and narrow band) to produce sky maps of stochastic gravitational-wave backgrounds and to search for gravitational waves from point sources. A spherical harmonic decomposition method is employed to look for gravitational-wave emission from spatially-extended sources. Neither technique found evidence of gravitational-wave signals. Hence we derive 95\% confidence-level upper limit sky maps on the gravitational-wave energy flux from broadband point sources, ranging from $F_{\alpha, \Theta} < {\rm (0.013 - 7.6)} \times 10^{-8} {\rm erg \, cm^{-2} \, s^{-1} \, Hz^{-1}},$ and on the (normalized) gravitational-wave energy density spectrum from extended sources, ranging from $\Omega_{\alpha, \Theta} < {\rm (0.57 - 9.3)} \times 10^{-9} \, {\rm sr^{-1}}$, depending on direction ($\Theta$) and spectral index ($\alpha$). These limits improve upon previous limits by factors of $2.9 - 3.5$. We also set 95\% confidence level upper limits on the frequency-dependent strain amplitudes of quasimonochromatic gravitational waves coming from three interesting targets, Scorpius X-1, SN 1987A and the Galactic Center, with best upper limits range from $h_0 < {\rm (1.7-2.1)} \times 10^{-25},$ a factor of $\geq 2.0$ improvement compared to previous stochastic radiometer searches.

Discovery of multiple p-mode pulsation frequencies in the roAp star, HD 86181

ABSTRACT We report the frequency analysis of a known roAp star, HD 86181 (TIC 469246567), with new inferences from Transiting Exoplanet Survey Satellite (TESS) data. We derive the rotation frequency to be νrot = 0.48753 ± 0.00001 d−1. The pulsation frequency spectrum is rich, consisting of two doublets and one quintuplet, which we interpret to be oblique pulsation multiplets from consecutive, high-overtone dipole, quadrupole, and dipole modes. The central frequency of the quintuplet is 232.7701 d−1 (2.694 mHz). The phases of the sidelobes, the pulsation phase modulation, and a spherical harmonic decomposition all show that the quadrupole mode is distorted. Following the oblique pulsator model, we calculate the rotation inclination, i, and magnetic obliquity, β, of this star, which provide detailed information about the pulsation geometry. The i and β derived from the best fit of the pulsation amplitude and phase modulation to a theoretical model, including the magnetic field effect, slightly differ from those calculated for a pure quadrupole, indicating the contributions from ℓ = 4, 6, 8,... are small. Non-adiabatic models with different envelope convection conditions and physics configurations were considered for this star. It is shown that models with envelope convection almost fully suppressed can explain the excitation at the observed pulsation frequencies.

Towards a robust estimation of orientation parameters between ICRF and Gaia celestial reference frames

ABSTRACT An analysis of the source position differences between very long baseline interferometry-based ICRF and Gaia-celestial reference frame catalogues is a key step in assessing their systematic errors and determining their mutual orientation. One of the main factors that limits the accuracy of determination of the orientation parameters between two frames is the impact of outliers. To mitigate this effect, a new method is proposed based on pixelization data over the equal-area cells, followed by median filtering of the data in each cell. After this, a new data set is formed, consisting of data points near-uniformly distributed over the sphere. The vector spherical harmonics decomposition is then applied to this data to finally compute the orientation parameters between ICRF and Gaia frames. To validate the proposed approach, a comparison was made of the ICRF3-SX and Gaia DR2 catalogues using several methods for outliers removal. The results of this work showed that the proposed method is practically insensitive to outliers and thus provides much more robust results of catalogues comparison than the methods used so far. This conclusion was confirmed by analogous test comparison of the Gaia DR2 and OCARS catalogues.

Application of improved EEMD algorithm in Low Carbon HVDC transmission line detection

In electronic detection and signal processing, the convenient and practical EEMD algorithm is generally selected. In this paper, the problems of the algorithm in some specific areas are improved. By adding white noise, the white noise is replaced by the white noise with opposite positive and negative amplitude, and the Schmidt orthogonalization theory is used to process each signal to eliminate the influence of modal function and improve the signal quality Anti mode aliasing ability, improve the level of multiple harmonic decomposition.

Perturbations of quasi-Newtonian universes in scalar–tensor gravity

In this contribution, we consider the well-known equivalence between [Formula: see text] gravity and Brans–Dicke-type scalar–tensor theories to study the evolution of scalar cosmological perturbations for a class of shear-free cosmological dust models with irrotational fluid flows. We use the [Formula: see text] covariant formalism to present the covariant linearized evolution and constraint equations. We then derive the integrability conditions describing a consistent evolution of the linearized field equations of quasi-Newtonian universes in the modified (scalar–tensor) theory of gravity. Finally, we derive the evolution equations for the density and velocity perturbations of the quasi-Newtonian universe. We apply the harmonic decomposition and explore the behavior of the matter density contrast by considering [Formula: see text] toy models. The growth of the matter density contrast for both short- and long-wavelength modes has been examined by applying certain assumptions of the initial conditions. We then apply the so-called quasi-static approximation to obtain exact solutions on small scales, but the results show that this approximation is not applicable here. Moreover, any small deviation from general relativity and any small change in the initial conditions of the perturbations causes huge orders-of-magnitude deviations from limiting general relativistic results, potentially putting constraints on the modified theory in the quasi-Newtonian cosmologies treatment. Our current work differs from other works in the literature, in that it is the first such work to show quasi-Newtonian cosmologies are unstable to linearized perturbations in modified gravity.

Harmonic cross-correlation decomposition for multivariate time series.

We introduce harmonic cross-correlation decomposition (HCD) as a tool to detect and visualize features in the frequency structure of multivariate time series. HCD decomposes multivariate time series into spatiotemporal harmonic modes with the leading modes representing dominant oscillatory patterns in the data. HCD is closely related to data-adaptive harmonic decomposition (DAHD) [Chekroun and Kondrashov, Chaos 27, 093110 (2017)10.1063/1.4989400] in that it performs an eigendecomposition of a grand matrix containing lagged cross-correlations. As for DAHD, each HCD mode is uniquely associated with a Fourier frequency, which allows for the definition of multidimensional power and phase spectra. Unlike in DAHD, however, HCD does not exhibit a systematic dependency on the ordering of the channels within the grand matrix. Further, HCD phase spectra can be related to the phase relations in the data in an intuitive way. We compare HCD with DAHD and multivariate singular spectrum analysis, a third related correlation-based decomposition, and we give illustrative applications to a simple traveling wave, as well as to simulations of three coupled Stuart-Landau oscillators and to human EEG recordings.

Machine learning for computationally efficient electrical loads estimation in consumer washing machines

Estimating the wear of the single electrical parts of a home appliance without resorting to a large number of sensors is desirable for ensuring a proper level of maintenance by the manufacturers. Deep learning techniques can be effective tools for such estimation from relatively poor measurements, but their computational demands must be carefully considered, for the actual deployment. In this work, we employ one-dimensional Convolutional Neural Networks and Long Short-Term Memory networks to infer the status of some electrical components of different models of washing machines, from the electrical signals measured at the plug. These tools are trained and tested on a large dataset (502 washing cycles approx 1000 h) collected from four different washing machines and are carefully designed in order to comply with the memory constraints imposed by available hardware selected for a real implementation. The approach is end-to-end; i.e., it does not require any feature extraction, except the harmonic decomposition of the electrical signals, and thus it can be easily generalized to other appliances.

InShaDe: Invariant Shape Descriptors for visual 2D and 3D cellular and nuclear shape analysis and classification

We present a shape processing framework for visual exploration of cellular nuclear envelopes extracted from microscopic images arising in histology and neuroscience. The framework is based on a novel shape descriptor of closed contours in 2D and 3D. In 2D, it relies on a geodesically uniform resampling of discrete curves to compute unsigned curvatures at vertices and edges based on discrete differential geometry. Our descriptor is, by design, invariant under translation, rotation, and parameterization. We achieve the latter invariance under parameterization shifts by using elliptic Fourier analysis on the resulting curvature vectors. Uniform scale-invariance is optional and is a result of scaling curvature features to z-scores. We further augment the proposed descriptor with feature coefficients obtained through sparse coding of the extracted cellular structures using K-sparse autoencoders. For the analysis of 3D shapes, we compute mean curvatures based on the Laplace-Beltrami operator on triangular meshes, followed by computing a spherical parameterization through mean curvature flow. Finally, we compute the Spherical Harmonics decomposition to obtain invariant energy coefficients. Our invariant descriptors provide an embedding into a fixed-dimensional feature space that can be used for various applications, e.g., as input features for deep and shallow learning techniques or as input for dimension reduction schemes to provide a visual reference for clustering shape collections. We demonstrate the capabilities of our framework in the context of visual analysis and unsupervised classification of 2D histology images and 3D nuclear envelopes extracted from serial section electron microscopy stacks.

Cosmological perturbations in f(G) gravity

We explore cosmological perturbations in a modified Gauss–Bonnet [Formula: see text] gravity, using a [Formula: see text] covariant formalism. In such a formalism, we define gradient variables to get perturbed linear evolution equations. We transform these linear evolution equations into ordinary differential equations using a spherical harmonic decomposition method. The obtained ordinary differential equations are time-dependent and then transformed into redshift-dependent. After these transformations, we analyze energy-density perturbations for two fluid systems, namely, for a Gauss–Bonnet field-dust system and for a Gauss–Bonnet field-radiation system for three different pedagogical [Formula: see text] models: trigonometric, exponential and logarithmic. For the Gauss–Bonnet field-dust system, energy-density perturbations decay with increase in redshift for all the three models. For the Gauss–Bonnet field-radiation system, the energy-density perturbations decay with increase in redshift for all of the three [Formula: see text] models for long wavelength modes whereas for short wavelength modes, the energy-density perturbations decay with increasing redshift for the logarithmic and exponential [Formula: see text] models and oscillate with decreasing amplitude for the trigonometric [Formula: see text] model.

Statistical exploration of halo anisotropic clustering and intrinsic alignments with the mass-Peak Patch algorithm

ABSTRACT The anisotropy or triaxiality of massive dark matter haloes largely defines the structure of the cosmic web, in particular the filaments that join the haloes together. Here, we investigate such oriented correlations in mass-Peak Patch halo catalogues by using the initial strain tensor of spherical proto-halo regions to orient the haloes. To go beyond the spherically averaged two-point correlation function of haloes, we use oriented stacks to compute oriented two-point correlations: we explicitly break isotropy by imposing a local frame set by the strain tensor of the reference halo before stacking neighbouring haloes. Beyond the exclusion zone of the reference halo, clustering is found to be strongly enhanced along the major direction of the strain tensor as expected. This anisotropic clustering of haloes along filaments is further quantified by using a spherical harmonics decomposition. Furthermore, we compute the evolution of cluster-scale halo principal directions relative to those of their neighbours and show that there are strong correlations extending up to very large scales. In order to provide calculations more suitable to observational confrontations, we also utilize 2D projected versions of some equivalent correlation functions. Finally, we show that the multipole structure of the mass-peak patch halo’s anisotropic clustering can be qualitatively captured in an analytic treatment based on peak theory. Though highly informative, giving the same qualitative features as the oriented correlations found from the simulation catalogue, analytic evaluation involves extensive use of Monte Carlo methods, which is also what the simulated catalogue uses, taking into account as they do the adaptive nature of the mass-peak patch mass hierarchy and all non-local complexities associated with the exclusion of smaller haloes overlapping with larger ones: there is no substitute for the mass-Peak Patch simulation-based determination of oriented and anisotropic correlations.

Continuous anisotropic damage as a twin modelling of discrete bi-dimensional fracture

In this contribution, the use of discrete simulations to formulate an anisotropic damage model is investigated. It is proposed to use a beam–particle model to perform numerical characterization tests. Indeed, this discrete model explicitly describes cracking by allowing displacement discontinuities and thus capture crack induced anisotropy of quasi-brittle materials such as concrete. Through 2D discrete simulations, the evolution of the effective elasticity tensor for various loading tests, up to failure, is obtained. The analysis of these tensors through bi-dimensional harmonic decomposition is then performed to estimate the tensorial damage evolution. It is shown in a quantitative manner that a second order –instead of a fourth order– damage tensor is sufficient in practice, even when the micro-cracks are strongly interacting. As a by-product of present work we obtain an upper bound of the distance to the orthotropic symmetry class of bi-dimensional elasticity.

Sequential dual sided SPIF using low order geometry reconstruction

Harmonic decomposition is an analytical technique that is able to express a manifold surface as the sum of a number of simple surface harmonic components. By reconstructing the initial geometry using a reduced number of components, a similar surface is obtained with a lower level of geometric detail. Because small features are filtered out and the resulting surface lies equal parts above and below the original surface, a tailored multi-step SPIF (Single Point Incremental Forming) processing strategy can be devised. This sequential SPIF strategy uses three processing passes to form a workpiece. The first step is a regular SPIF operation using a conventional toolpath strategy to form the reduced geometry. Two finishing steps are then needed, one from the same side to form the smaller features that lies deeper than the reduced geometry and one backwards pass from the other side of the sheet. To add features that need to be shallower than the reduced geometry, the part is flipped around. The used sequence of these finishing steps and the toolpath strategy used significantly influence the final part accuracy and surface quality. The advantages and disadvantages of four of these combined strategies are examined and compared to regular SPIF.

1 + 3 covariant perturbations in power-law f(R) gravity

We applied the 1+3 covariant approach around the Friedmann–Lemaître–Robertson–Walker (FLRW) background, together with the equivalence between f(R) gravity and scalar-tensor theory to study cosmological perturbations. We defined the gradient variables in the 1 + 3 covariant approach which we used to derive a set of evolution equations. Harmonic decomposition was applied to partial differential equations to obtain ordinary differential equations used to analyse the behavior of the perturbation quantities. We focused on dust dominated area and the perturbation equations were applied to background solution of alpha R+beta R^{n} model, n being a positive constant. The transformation of the perturbation equations into redshift dependence was done. After numerical solutions, it was found that the evolution of energy-density perturbations in a dust-dominated universe for different values of n decays with increasing redshift.

Extracorporeal circulation during on-pump cardiac surgery: An evaluation of the energy equivalent pressure index based on waveforms decomposition in harmonics.

The use of pulsatile perfusion instead of nonpulsatile perfusion during cardiopulmonary bypass continues to be a source of debate. The disagreements among the conclusions of the published studies may be due to different factors: differences in the type of patients included in the studies, differences in the protocol of the studies, and difficulty to quantify the pulsatility of the flow. In the present paper, we propose a quantitative evaluation of Shepard's energy equivalent pressure index, based on the harmonic decomposition of the physiological aortic pressure and flow rate signal. It is thus demonstrated that the surplus energy provided by pulsatile flow remains moderate (of order 10mmHg), but that it can be improved by changing the relative shapes of the pressure and flow waves.

Experimental and numerical investigation of a surface-fixed horizontal porous wave barrier

Experimental and numerical investigations were conducted to study the performance of a surface-fixed horizontal porous wave barrier in regular waves. The characteristics of the reflection and transmission coefficients, energy dissipation, and vertical wave force were examined versus different porosities of the barrier. Numerical simulations based on 3D Reynolds Averaged Navier-Stokes equations with standard low-Re k-e turbulent closure and volume of fluid approach were accomplished and compared with the experimental results conducted in a 2D wave tank. Experimental measurements and numerical simulations were shown to be in satisfactory agreement. The qualitative wave behavior propagating over a horizontal porous barrier such as wave run-up, wave breaking, air entrapment, jet flow, and vortex generation was reproduced by CFD computation. Through the discrete harmonic decomposition of the vertical wave force on a wave barrier, the nonlinear characteristics were revealed quantitatively. It was concluded that the surface-fixed horizontal barrier is more effective in dissipating wave energy in the short wave period region and more energy conversion was observed from the first harmonic to higher harmonics with the increase of porosity. The present numerical approach will provide a predictive tool for an accurate and efficient design of the surface-fixed horizontal porous wave barrier.

New inverse wavelet transform method with broad application in dynamics

Extracting multi-scale models from system identification of stationary or nonstationary measured signals (e.g., time series) is of great importance in engineering and the applied sciences. We propose a new computational method for harmonic analysis and decomposition of signals based on the inverse wavelet transform and demonstrate its efficacy in diverse areas in dynamics. The wavelet transform is a linear transformation of a signal measured in the temporal/spatial domain to the time–frequency/space-wavenumber domain and applies to stationary and nonstationary measurements. The new method is based on a numerical inverse wavelet transform and yields decomposition of the measured signal in terms of its dominant harmonic components. First, we formulate the analytical continuous inverse wavelet transform in a way that is suitable for computational implementation. Then, taking as example a general measured signal in the time domain, (i) we numerically compute its numerical wavelet transform spectrum, (ii) define a set of “harmonic regions” in the wavelet spectrum containing the dominant harmonics to be inverted and studied, and (iii) by numerically inverse wavelet transforming each of the harmonic regions separately, obtain the respective decomposed harmonics in the time domain. Note that, by construction, the superposition of all decomposed harmonics reconstructs the original signal. Next, we demonstrate the efficacy of the method with some examples. We start with an artificial signal with prescribed harmonic components to highlight the method and its accuracy. Then, we show applicability of the method to system identification, by applying it to the modal analysis of a system of linearly coupled oscillators with closely spaced modes. Lastly, we show how the new method enables quantification of the energy captured by each of the decomposed components (harmonics) in the response of a strongly nonlinear system. To this end, a single degree of freedom geometrically nonlinear oscillator is considered, and the method is used to quantify nonlinear energy “scattering” in its frequency domain. These examples hint at the broad applicability of the new method to diverse areas of signal processing and dynamics, including discrete and continuous dynamical systems with strongly (and even non-smooth) nonlinearities.

High-order harmonic generation in graphene: Nonlinear coupling of intraband and interband transitions

We investigate high-order harmonic generation (HHG) in graphene with a quantum master equation approach. The simulations reproduce the observed enhancement in HHG in graphene under elliptically polarized light [N. Yoshikawa et al, Science 356, 736 (2017)]. On the basis of a microscopic decomposition of the emitted high-order harmonics, we find that the enhancement in HHG originates from an intricate nonlinear coupling between the intraband and interband transitions that are respectively induced by perpendicular electric field components of the elliptically polarized light. Furthermore, we reveal that contributions from different excitation channels destructively interfere with each other. This finding suggests a path to potentially enhance the HHG by blocking a part of the channels and canceling the destructive interference through band-gap or chemical potential manipulation.

Possible risk resulting from the recent decay of the dipolar component of the terrestrial magnetic field

In this study, we investigated the geomagnetic ground observatory data from 1980 to 2011 collected from World Data Center from 134 stations. To analyze the data we have applied spherical harmonic decomposition to obtain components associated with the Earth’s main magnetic field and to calculate how the Earth’s dipole was varying in the aforementioned recent 31-year period. There is a visible ~ 2.3% decay of the dipole magnetic field of the Earth. We note that the present-day value of the magnetic dipole intensity is the lowest one in the history of modern civilization and that further drop of this value may pose a risk for different domains of our life.