Harmonic Decomposition Research Articles

Crustal anisotropy beneath northeastern Tibetan Plateau from the harmonic decomposition of receiver functions

SUMMARY A uniformly spaced linear transect through the northeastern Tibetan Plateau was constructed using 54 stations from ChinaArray Phase II. We used a set of colocated earthquakes to form receiver function beams that were then used to construct a 2-D image of main converting boundaries in our region and to investigate lateral changes in main impedance contrasts along the transect. The image revealed obvious mid-crustal low-velocity zones beneath the Qilian Orogen and the Alxa Block. We developed a new procedure that uses harmonically decomposed receiver functions to characterize seismic anisotropy, and that can determine both the orientations of symmetry axes and their type (fast or slow). We tested our technique on a number of synthetic models, and subsequently applied it to the data from the transect. We found that: (1) within the upper crust the orientations of slow symmetry axes are nearly orthogonal to the strike directions of faults, and thus anisotropy is likely caused by the shape preferred orientation of fluid-saturated cracks or fractures and (2) together with the low-velocity zones revealed from receiver functions stacks, anisotropic layers in the middle-to-lower crust could be explained by the crustal channel flow that was proposed for this region by previous studies. The shear within the boundary layers of crustal flow forms anisotropy with symmetry axes parallel to the flow direction.

Experimental investigation on the power capture of an oscillating wave surge converter in unidirectional waves

The aim of this paper is the experimental characterization of the capture width ratio (CWR) and response amplitude operator (RAO) of a 1:10 physical model of an oscillating wave surge converter (OWSC), under unidirectional regular and irregular waves. The effects of hydraulic power take-off (PTO) system are explicitly taken in consideration. A mathematical model is proposed to describe the PTO damping as a function of the angular velocity of the flap. The harmonic decomposition of free-surface elevation, angular velocity of the flap and pressure in the PTO system demonstrates that these signals are dominated by their linear component. However, the signals also present some important higher-order frequency components. To predict the CWR of the OWSC under irregular waves the nonlinear output frequency response functions are considered as the extension of the RAO to the nonlinear case. It is shown and discussed that the PTO system, wave frequency and height have a significant influence on the CWR and RAO. The RAO curve for irregular waves does not exhibit a well-defined peak, showing a limited variation in a broadband. A weak correlation between CWR and RAO was found, i.e., the maximum CWR does not occur for the maximum value of RAO.

Super-sample tidal modes on the celestial sphere

The super-sample tidal effect carries information on long-wavelength fluctuations that we cannot measure directly. It arises from the mode-coupling between short-wavelength and long-wavelength perturbations beyond a finite region of a galaxy survey and violates statistical isotropy of observed galaxy power spectra. In this paper, we propose the use of bipolar spherical harmonic (BipoSH) decomposition formalism to characterize statistically anisotropic power spectra. Using the BipoSH formalism, we perform a comprehensive study of the effect of the super-sample tides on measurements of other cosmological distortions such as the redshift-space distortion (RSD) and Alcock-Paczy\'{n}ski (AP) effects by means of the Fisher information matrix formalism. We find that the BipoSH formalism can break parameter degeneracies among the super-sample tidal, RSD and AP effects, indicating that the super-sample tides have little impact on the measurements of the RSD and AP effects. We also show that the super-sample tides are detectable with an accuracy better than the $\Lambda$CDM prediction without impairing the accuracy of measurements of other anisotropies assuming a SPHEREx-like galaxy survey.

Analysis of phased array antenna system via spherical harmonics decomposition

Spherical phased-array antenna has hemispherical scan coverage for transmission of satellite imagery data at high bit rates to the ground station. Different conditions have been assigned based on their effects on the overall performance of the communication link. Reliability analysis consequently becomes an important requirement of any system onboard a satellite. Although the phased array and some connected algorithms have been presented in the literatures; however, there is no extensive systematic and theoretical performance evaluation framework formulated. In this work, the authors present new method that efficiently and effectively analyses the statistical performance of phased-array antenna systems using spherical harmonics expansion approach. Specifically, the errors caused by limited antenna element number, measurement noise of antenna, imprecision in phased-array antenna elements positioning, and spatial aliasing are systematically and theoretically benchmarked via simulations. Spatial sampling mechanisms for the positioning of array antenna elements on a sphere are presented and evaluated. This study proposed design guide and analysis for spherical phased-array antenna systems, and antenna arrays that do not employ spherical harmonics explicitly.

K-PHD Tek-Ton Frekans Kestiricisinin Bir Kesin Değişinti İfadesi: Rasgele Faz Durumu

The k-PHD method, a generalization of the well-known Pisarenko harmonic decomposition (PHD) method, is considered for frequency estimation of a single real random-phased sinusoid in noise. With the use of a simple variance analysis technique, an exact expression of the k-PHD frequency variance is derived. An approximate k-PHD variance formula for sufficiently large data lengths and signal-to-noise ratios is also given. Computer simulations are included to validate the theoretical development.

On data-driven augmentation of low-resolution ocean model dynamics

The problem of augmenting low-resolution ocean circulation models with the information extracted from the data relevant to the unresolved subgrid processes is addressed. A highly nonlinear model of eddy-resolving oceanic circulation – quasigeostrophic wind-driven double gyres – is considered. The model solutions are characterized by a vigorous dynamic coupling between the resolved large-scale and small-scale (eddy) flow features. This solution provides the data for augmenting the low-resolution model with the same configuration. The eddy forcing field, which contains the essential information about coupling between the large and eddy scales, is obtained, modified, coarse-grained and added to augment the low-resolution model. The implemented modification involves novel data-adaptive harmonic decomposition analysis and dynamical constraining based on the low-resolution nonlinear advection operator. The resulting augmentation of the low-resolution model significantly improves the solution, including its time-mean circulation and low-frequency variability. This result also paves the way for a systematic data-driven emulation of unresolved and under-resolved scales of motion.

Complete multipolar description of reflection and transmission across a metasurface for perfect absorption of light.

Relating the electromagnetic scattering and absorption properties of an individual particle to the reflection and transmission coefficients of a two-dimensional material composed of these particles is a crucial concept that has driven both fundamental and applied physics. It is at the heart of both the characterization of material properties as well as the phase and amplitude engineering of a wave. Here we propose a multipolar description of the reflection and transmission coefficients across a monolayer of particles using a vector spherical harmonic decomposition. This enables us to provide a generalized condition for perfect absorption which occurs when both the so-called generalized Kerker condition is reached and when the superposition of odd and even multipoles reaches a critical value. Using these conditions, we are able to propose two very different designs of two-dimensional materials that perfectly absorb a plane electromagnetic wave under normal incidence. One is an infinite array of silica microspheres that operates at mid-infrared frequencies, while the other is an infinite array of germanium nano-clusters that operates at visible frequencies. Both systems operate in a deeply multipolar regime. Our findings are important to the metamaterials and metasurfaces communities who design materials mainly restricted to the dipolar behavior of individual resonators, as well as the self-assembly and nanochemistry communities who separate the individual particle synthesis from the materials assembly.

Tesseral superconducting shim coil design with quasi-saddle geometry for use in high-field magnet system.

Superconducting shim coils are frequently used in high-field magnetic resonance imaging or nuclear magnetic resonance system for their high sensitivity and shimming strength. The design of superconducting shim coils is based on the spherical harmonic decomposition, and each shim coil is normally dedicated for correcting one specific harmonic component. Conventional superconducting shim coil with a saddle loop has observable winding error near the corner, which gives rise to arc transformation when winding layer by layer. Simulation analysis shows that the arc corner transformation will induce the magnetic field deviation by more than double of the theoretical design ±1%, which may be up to ±3% after real winding. An improved shim coil design method with a quasisaddle geometry was proposed to correct the winding error. With the consideration of both the rounded corner of the saddle loop and the arc side, the new design offers the magnetic field deviation within ±1%. In addition to reducing the winding error, the proposed design also facilitates the winding process.

Robust Time-Domain Broadband Modal Beamforming for Circular Arrays

Modal beamformer for circular arrays, which is based on a circular harmonics decomposition of the sound field, is flexible for use because it is decoupled from the sensor positions. A time-domain implementation of broadband modal beamformer for circular arrays is presented in this paper. This beamforming structure consists of a real-valued circular harmonics transform and steering unit, followed by a pattern generation unit implemented by a set of real-valued finite impulse response (FIR) filters. The beam pattern is controlled through the FIR filters' tap weights, whose design is decoupled from beam pattern steering and can be formulated as a constrained optimization problem that can be solved efficiently. Multiple array performance measures such as array gain, sensitivity, mainlobe, and sidelobes can be taken into consideration in beamformer design, which is very important in practical applications. Simulation and experimental results are presented to illustrate the effectiveness of the proposed time-domain broadband modal beamforming approach. Performance comparisons between the proposed and the classical methods in weight coefficients, beam patterns, and quality of audio output are also provided. The results show that the proposed approach possesses high robustness and can achieve good frequency invariance.

A new proof of the Hardy–Rellich inequality in any dimension

The Hardy-Rellich inequality in the whole space with the best constant was firstly proved by Tertikas and Zographopoulos in Adv. Math. (2007) in higher dimensions N ⩾ 5. Then it was extended to lower dimensions N ∈ {3, 4} by Beckner in Forum Math. (2008) and Ghoussoub-Moradifam in Math. Ann. (2011) by applying totally different techniques.In this note, we refine the method implemented by Tertikas and Zographopoulos, based on spherical harmonics decomposition, to give an easy and compact proof of the optimal Hardy–Rellich inequality in any dimension N ⩾ 3. In addition, we provide minimizing sequences which were not explicitly mentioned in the quoted papers in lower dimensions N ∈ {3, 4}, emphasizing their symmetry breaking. We also show that the best constant is not attained in the proper functional space.

Estimating Four-Dimensional Internal Wave Spectrum in the Northern South China Sea

AbstractInternal waves can transfer energy from large-scale to microscale processes; however, the spectra of these waves remain poorly known. A method that combines modal harmonic decomposition and maximum-likelihood method is proposed in this study to estimate four-dimensional internal wave spectrum using limited mooring observations. Using this method, a four-dimensional internal wave spectrum was obtained for the first time based on the mooring measurements collected during the South China Sea (SCS) Internal Wave Experiment in July 2014. The spectrum was then validated by comparing with the spectrum based on Fourier analysis and with the modified Garrett–Munk internal wave spectrum, respectively. The power of the internal wave spectrum decreased obviously with increasing frequency and wavenumber, with a falloff rate of ω−2 beyond tidal frequencies, and with falloff rates of and for horizontal and vertical wavenumbers, respectively. In addition, at a fixed frequency and vertical wavenumber, the propagation direction and phase speed of internal waves can be obtained through the four-dimensional spectrum. In summary, we verified the feasibility of estimating four-dimensional internal wave spectrum using limited mooring observations in this study, and the method we proposed should be applicable to other regions where such mooring observations are available.

DC-side harmonic analysis and DC filter design in hybrid HVDC transmission systems

Hybrid high-voltage direct current (HVDC) transmission is a new transmission structure that uses line commutated converter (LCC) at the rectifier side and voltage source converter (VSC) at the inverter side. This study proposes a method to calculate DC side harmonic currents of hybrid HVDC transmission systems and design filters to mitigate DC-side low-frequency voltage pulsations. First, an equivalent harmonic analysis model of modular multilevel converter (MMC) is deduced according to the relationship among different harmonic components. The harmonic voltage expressions at DC side of 12-pulse LCC are deduced by combining the three-pulse harmonic voltage source model and FFT decomposition. On this basis, the expressions for harmonic currents are derived by multiplying the harmonic voltage transfer function and transfer admittance equation of DC network and are solved for the inverse Laplace transform. Two different types of DC filters are designed according to the requirements of DC filter on decaying specific order harmonic currents. The influence of partial element decay on the transfer characteristic of filter is analyzed, and appropriate partial component dissipation parameters are applied. The decay pole near the harmonic resonance frequency is configured to achieve reasonable DC-filtering results. Simulation results of a hybrid HVDC transmission system demonstrate the effectiveness of the proposed analysis model and filter design method.

Effect of filtered feedback on birhythmicity: Suppression of birhythmic oscillation.

The birhythmic oscillation, generally known as birhythmicity, arises in a plethora of physical, chemical, and biological systems. In this paper we investigate the effect of filtered feedback on birhythmicity as both are relevant in many living and engineering systems. We show that the presence of a low-pass filter in the feedback path of a birhythmic system suppresses birhythmicity and supports monorhythmic oscillations depending on the filtering parameter. Using harmonic decomposition and energy balance methods we determine the conditions for which birhythmicity is removed. We carry out a detailed bifurcation analysis to unveil the mechanism behind the quenching of birhythmic oscillations. Finally, we demonstrate our theoretical findings in analog simulation with electronic circuit. This study may have practical applications in quenching birhythmicity in several biochemical and physical systems.

Three-dimensional wave-domain acoustic contrast control using a circular loudspeaker array.

This paper proposes a three-dimensional wave-domain acoustic contrast control method to reproduce a multizone sound field using a circular loudspeaker array. In this method, sound field analysis is based on spherical harmonic decomposition, and the loudspeaker weights are obtained by maximizing the acoustic energy contrast between the predefined bright zone and dark zone. Simulation results show that the proposed method provides good multizone separation performance over a large spatial region and requires lower-order spherical harmonics, resulting in a much lower number of microphones required to measure the acoustic transfer functions.

Shape analysis of 3D nanoscale reconstructions of brain cell nuclear envelopes by implicit and explicit parametric representations

Shape analysis of cell nuclei is becoming increasingly important in biology and medicine. Recent results have identified that large variability in shape and size of nuclei has an important impact on many biological processes. Current analysis techniques involve automatic methods for detection and segmentation of histology and microscopy images, but are mostly performed in 2D. Methods for 3D shape analysis, made possible by emerging acquisition methods capable to provide nanometric-scale 3D reconstructions, are still at an early stage, and often assume a simple spherical shape. We introduce here a framework for analyzing 3D nanoscale reconstructions of nuclei of brain cells (mostly neurons), obtained by semiautomatic segmentation of electron micrographs. Our method considers two parametric representations: the first one customizes the implicit hyperquadricsformulation and it is particularly suited for convex shapes, while the latter considers a spherical harmonics decomposition of the explicit radial representation. Point clouds of nuclear envelopes, extracted from image data, are fitted to the parameterized models which are then used for performing statistical analysis and shape comparisons. We report on the analysis of a collection of 121 nuclei of brain cells obtained from the somatosensory cortex of a juvenile rat.

Advances in Concurrent Computing for Digital Stochastic Measurement Simulation

This paper introduces a concurrent computing technique for the acceleration of digital stochastic measurement simulations. The digital stochastic measurement presents an advanced methodology based on the specific parallel hardware structure, utilized for an orthogonal transformation calculus/decomposition. Methodology is analyzed in detail, starting from the very basic idea, toward recent references, covering main research directions and trends. An oversampling nature of the evaluated digital stochastic measurement, along with demanding arithmetic requirements, implies exhausting simulation complexity. As a test case, several typical power grid signals were harmonically analyzed through a discrete Fourier transformation based on the proposed methodology. A harmonic decomposition was simulated with several levels of computing concurrency. Through all the simulated scenarios main success criterion was model accuracy, while the parameter used for selection of the optimal simulation computing technique was the overall calculus speed. Final results exposed thread pool computing technique as an optimal simulation platform.

Quaternionic spherical harmonics and a sharp multiplier theorem on quaternionic spheres

A sharp $L^p$ spectral multiplier theorem of Mihlin--H\"ormander type is proved for a distinguished sub-Laplacian on quaternionic spheres. This is the first such result on compact sub-Riemannian manifolds where the horizontal space has corank greater than one. The proof hinges on the analysis of the quaternionic spherical harmonic decomposition, of which we present an elementary derivation.

Harmonic decomposition and analysis of running gait

Harmonic decomposition and analysis of running gait

On the saturation of acoustic mode frequencies at high solar activity

ABSTRACT Acoustic mode frequencies obtained by applying spherical harmonic decomposition to HMI, MDI, and GONG observations were analysed throughout the solar cycle. Evidence of a deviation from a linear relation with solar radio flux was found indicating a saturation effect at high solar activity. The Gompertz model, which is one of the most frequently used sigmoid functions to fit growth data, is used. It is shown that its fitting to MDI and GONG data are statistically significant and a median saturation of 400 sfu is estimated. This saturation level is 50 per cent larger than any obtained in the last century, hence the small effect observed in the minimum-to-maximum frequency shift. However, as shown here, it should not be disregarded.

Calculation of Stall Margin Enhancement With Micro-Tip Injection in an Axial Compressor

Although steady micro-injection is experimentally validated as an attractive method in improving the stall margin of axial compressors, up to now a fast prediction of stall boundary remains some way off. This investigation is to propose such a prediction model. A flow stability model is developed to further consider the effect of high-speed micro-injection. After the base flow field is calculated by steady computational fluid dynamics simulation, a body force model is applied to reproduce the effect of blade on the flow turning and loss. A group of homogeneous equations are obtained based on linearized Navier–Stokes equations and harmonic decomposition of small flow disturbance. The stall onset point can be judged by the imaginary part of the resultant eigenvalue. After the existing experimental results are summarized, an unsteady numerical simulation reveals that the computed characteristics and radial profile of pressure rise coefficient are almost unchanged. The unsteady response of compressor to the micro-injection is preliminarily verified based on the observation of the disturbed spillage of tip leakage flow. It is verified that this approach can provide a qualitative assessment of stall point with acceptable computational cost. Both high injection velocity and short axial gap between injector and rotor leading edge are beneficial for the stall margin extension. These theoretical findings agree well with experimental measurements. It is inferred that the spillage of tip clearance flow, which is inward pushed by higher speed injection with shortened distance away from rotor, could lead to further stable flow field.