Wave Shape Functions Research Articles

Ridge Detection for Nonstationary Multicomponent Signals With Time-Varying Wave-Shape Functions and its Applications

Ridge Detection for Nonstationary Multicomponent Signals With Time-Varying Wave-Shape Functions and its Applications

Time Variant Wave-Signal-Amplitude Trigonometry Regression of Latitudes and Longitudes of the Belmullets of the Atlantic Ocean

This paper introduces time variant wave-signal-amplitude cosine and sine regression as an extension to wave signal Fourier function and Wave-Shape Function (WSF) model. A full-scale conditional characterization of the linear time variant wave-signal-amplitude cosine and sine model of cosine and sine function with random errors (ηi) was proposed. The associated regression coefficients were estimated via the Ordinary Least Square (OLS) technique, such that, the model wave signal, frequency, and phase were carved-out. In application to real life problem, the wave-signal-amplitude trigonometry model was applied to the real-time observations of the latitude and longitude of the wave buoys’ Belmullets of the Atlantic Ocean. The full-scale real-time observations of the wave climate are the time-variant significant wave height (in metre), peak wave (in oC) and sea temperature (in oC) from 2012 to 2022.

An Iterative Warping and Clustering Algorithm to Estimate Multiple Wave-Shape Functions From a Nonstationary Oscillatory Signal

Nonsinusoidal oscillatory signals are everywhere. In practice, the nonsinusoidal oscillatory pattern, modeled as a 1-periodic wave-shape function (WSF), might vary from cycle to cycle. When there are finite different WSFs, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$s_{1},\ldots,s_{K}$</tex-math></inline-formula> , so that the WSF jumps from one to another suddenly, the different WSFs and jumps encode useful information. We present an iterative warping and clustering algorithm to estimate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$s_{1},\ldots,s_{K}$</tex-math></inline-formula> from a nonstationary oscillatory signal with time-varying amplitude and frequency, and hence the change points of the WSFs. The algorithm is a novel combination of time-frequency analysis, singular value decomposition entropy and vector spectral clustering. We demonstrate the efficiency of the proposed algorithm with simulated and real signals, including the voice signal, arterial blood pressure, electrocardiogram and accelerometer signal. Moreover, we provide a mathematical justification of the algorithm under the assumption that the amplitude and frequency of the signal are slowly time-varying and there are finite change points that model sudden changes from one wave-shape function to another one.

High frequency homogenization for a one-dimensional acoustic black hole lattice

A hybrid theoretical framework was developed to understand the wave propagation properties of acoustic black hole (ABH) lattices, based on a combination of WKB theory and the high-frequency homogenization method. Explicit approximate expressions were obtained for the associated dispersion curves. Firstly, a matrix formulation to model wave propagation in a one-dimensional lattice was developed, where the wave components of the flexural displacement were described as WKB solutions of the flexural wave equation. The standing wave frequencies and the corresponding wave shape functions required by the high-frequency homogenization analysis were obtained accordingly. Secondly, the effective dispersion relation of the ABH lattice was obtained analytically by utilizing high-frequency homogenization. Based on the dispersion results obtained from the hybrid framework, the group velocity and hence the modal density of the ABH lattice were obtained analytically. Finally, the proposed technique was validated through several numerical examples.

Wave-shape function model order estimation by trigonometric regression

The adaptive non-harmonic (ANH) model is a powerful tool to compactly represent oscillating signals with time-varying amplitude and phase, and non-sinusoidal oscillating morphology. Given good estimators of instantaneous amplitude and phase we can construct an adaptive model, where the morphology of the oscillation is described by the wave-shape function (WSF), a 2π-periodic more general periodic function. In this paper, we address the problem of estimating the number of harmonic components of the WSF, a problem that remains underresearched, by adapting trigonometric regression model selection criteria into this context. We study the application of these criteria, originally developed in the context of stationary signals, to the case of signals with time-varying amplitudes and phases. We then incorporate the order estimation to the ANH model reconstruction procedure and analyze its performance for noisy AM-FM signals. Experimental results on synthetic signals indicate that these criteria enable the adaptive estimation of the waveform of non-stationary signals with non-sinusoidal oscillatory patterns, even in the presence of considerable amount of noise. We also apply our reconstruction procedure to the task of denoising simulated pulse wave signals and determine that the proposed technique performs competitively to other denoising schemes. We conclude this work by showing that our adaptive order estimation algorithm takes into account the interpatient waveform variability of the electrocardiogram (ECG) and respiratory signals by analyzing recordings from the Fantasia Database.

Decomposing Non-Stationary Signals With Time-Varying Wave-Shape Functions

Modern time series are usually composed of multiple oscillatory components, with time-varying frequency and amplitude contaminated by noise. The signal processing mission is further challenged if each component has an oscillatory pattern, or the wave-shape function, far from a sinusoidal function, and the oscillatory pattern is even changing from time to time. In practice, if multiple components exist, it is desirable to robustly decompose the signal into each component for various purposes, and extract desired dynamics information. Such challenges have raised a significant amount of interest in the past decade, but a satisfactory solution is still lacking. We propose a novel {\em nonlinear regression scheme} to robustly decompose a signal into its constituting multiple oscillatory components with time-varying frequency, amplitude and wave-shape function. We coined the algorithm {\em shape-adaptive mode decomposition (SAMD)}. In addition to simulated signals, we apply SAMD to two physiological signals, impedance pneumography and electroencephalography. Comparison with existing solutions, including linear regression, recursive diffeomorphism-based regression and multiresolution mode decomposition, shows that our proposal can provide an accurate and meaningful decomposition with computational efficiency.

Influence of Strain Rate and Waveshape on Environmentally-Assisted Cracking during Low-Cycle Fatigue of a 304L Austenitic Stainless Steel in a PWR Water Environment

In this paper, the low cycle fatigue resistance of a 304L austenitic stainless steel in a simulated pressurized water reactor (PWR) primary water environment has been investigated by paying a special attention to the interplay between environmentally-assisted cracking mechanisms, strain rate, and loading waveshape. More precisely, one of the prime interests of this research work is related to the consideration of complex waveshape signals that are more representative of solicitations encountered by real components. A detailed analysis of stress-strain relation, surface damage, and crack growth provides a preliminary ranking of the severity of complex, variable strain rate signals with respect to triangular, constant strain-rate signals associated with environmental effects in air or in PWR water. Furthermore, as the fatigue lives in PWR water environment are mainly controlled by crack propagation, the crack growth rates derived from striation spacing measurement and estimated from interrupted tests have been carefully examined and analyzed using the strain intensity factor range ΔKε. It is confirmed that the most severe signal with regards to fatigue life also induces the highest crack growth enhancement. Additionally two characteristic parameters, namely a threshold strain εth* and a time T*, corresponding to the duration of the effective exposure of the open cracks to PWR environment have been introduced. It is shown that the T* parameter properly accounts for the differences in environmentally-assisted growth rates as a function of waveshape.

Electrostatic self-cleaning gecko-like adhesives.

This paper describes the use of the electrostatic element of an electrostatic/gecko-like adhesive to repel dust particles, which have been shown to significantly affect adhesion and reliability. The result is a non-destructive, non-contact cleaning method that can be used in conjunction with other cleaning techniques, many of which rely on physical contact between the fibrillar adhesive and substrate. The paper focuses on experimental evaluation of the repulsion of 100 μm glass beads as a function of wave shape, frequency, phase number and electrode direction in relation to the gecko-like features. Results show that a two-phase square wave with the lowest practically feasible frequency can remove 100 μm glass beads from a directional gecko-like adhesive with up to 70% efficiency. Finally, using the optimized electrostatic cleaning properties, results show an approximately 25% recovery in shear stress on a rough glass for three contaminated directional gecko-like adhesives after contact with a dusty table.

Wave-Shape Function Analysis

We propose to combine cepstrum and nonlinear time–frequency (TF) analysis to study multiple component oscillatory signals with time-varying frequency and amplitude and with time-varying non-sinusoidal oscillatory pattern. The concept of cepstrum is applied to eliminate the wave-shape function influence on the TF analysis, and we propose a new algorithm, named de-shape synchrosqueezing transform (de-shape SST). The mathematical model, adaptive non-harmonic model, is introduced and the de-shape SST algorithm is theoretically analyzed. In addition to simulated signals, several different physiological, musical and biological signals are analyzed to illustrate the proposed algorithm.

Modeling the Pulse Signal by Wave-Shape Function and Analyzing by Synchrosqueezing Transform.

We apply the recently developed adaptive non-harmonic model based on the wave-shape function, as well as the time-frequency analysis tool called synchrosqueezing transform (SST) to model and analyze oscillatory physiological signals. To demonstrate how the model and algorithm work, we apply them to study the pulse wave signal. By extracting features called the spectral pulse signature, and based on functional regression, we characterize the hemodynamics from the radial pulse wave signals recorded by the sphygmomanometer. Analysis results suggest the potential of the proposed signal processing approach to extract health-related hemodynamics features.

Optimizing Estimates of Instantaneous Heart Rate from Pulse Wave Signals with the Synchrosqueezing Transform.

With recent advances in sensor and computer technologies, the ability to monitor peripheral pulse activity is no longer limited to the laboratory and clinic. Now inexpensive sensors, which interface with smartphones or other computer-based devices, are expanding into the consumer market. When appropriate algorithms are applied, these new technologies enable ambulatory monitoring of dynamic physiological responses outside the clinic in a variety of applications including monitoring fatigue, health, workload, fitness, and rehabilitation. Several of these applications rely upon measures derived from peripheral pulse waves measured via contact or non-contact photoplethysmography (PPG). As technologies move from contact to non-contact PPG, there are new challenges. The technology necessary to estimate average heart rate over a few seconds from a noncontact PPG is available. However, a technology to precisely measure instantaneous heat rate (IHR) from non-contact sensors, on a beat-to-beat basis, is more challenging. The objective of this paper is to develop an algorithm with the ability to accurately monitor IHR from peripheral pulse waves, which provides an opportunity to measure the neural regulation of the heart from the beat-to-beat heart rate pattern (i.e., heart rate variability). The adaptive harmonic model is applied to model the contact or non-contact PPG signals, and a new methodology, the Synchrosqueezing Transform (SST), is applied to extract IHR. The body sway rhythm inherited in the non-contact PPG signal is modeled and handled by the notion of wave-shape function. The SST optimizes the extraction of IHR from the PPG signals and the technique functions well even during periods of poor signal to noise. We contrast the contact and non-contact indices of PPG derived heart rate with a criterion electrocardiogram (ECG). ECG and PPG signals were monitored in 21 healthy subjects performing tasks with different physical demands. The root mean square error of IHR estimated by SST is significantly better than commonly applied methods such as autoregressive (AR) method. In the walking situation, while AR method fails, SST still provides a reasonably good result. The SST processed PPG data provided an accurate estimate of the ECG derived IHR and consistently performed better than commonly applied methods such as autoregressive method.

A Solver for Helmholtz System Generated by the Discretization of Wave Shape Functions

AbstractAn interesting discretization method for Helmholtz equations was introduced in B. Després [1]. This method is based on the ultra weak variational formulation (UWVF) and the wave shape functions, which are exact solutions of the governing Helmholtz equation. In this paper we are concerned with fast solver for the system generated by the method in [1]. We propose a new preconditioner for such system, which can be viewed as a combination between a coarse solver and the block diagonal preconditioner introduced in [13]. In our numerical experiments, this preconditioner is applied to solve both two-dimensional and three-dimensional Helmholtz equations, and the numerical results illustrate that the new preconditioner is much more efficient than the original block diagonal preconditioner.

Study on far field wave patterns and their characteristics of Havelock form green function

A new mathematical integral representation including five integrals about the far field wave shape function of Havelock form translating-pulsating source is obtained by performing variable substitution. Constant-phase curves and propagation wave patterns are investigated by applying stationary phase analysis method to the new representation. Some findings are summarized as follows: (1) when 0 0.25, besides three types of wave patterns such as a ring-faning wave pattern, a fan wave pattern and an inner V wave pattern, a new one called parallel wave pattern is also found which not only exists in the integrals about the ring-fan wave and fan wave, but also in the integrals whose interval is [0,γ] In addition, Characteristics about these parallel waves such as mathematical expressions, existence conditions, propagation directions and wave lengths are obtained, and cancellation relationships between these parallel waves are stated, which certificates the fact that there are no parallel waves existing in the far field.

Instantaneous frequency and wave shape functions (I)

Although one can formulate an intuitive notion of instantaneous frequency, generalizing “frequency” as we understand it in e.g. the Fourier transform, a rigorous mathematical definition is lacking. In this paper, we consider a class of functions composed of waveforms that repeat nearly periodically, and for which the instantaneous frequency can be given a rigorous meaning. We show that Synchrosqueezing can be used to determine the instantaneous frequency of functions in this class, even if the waveform is not harmonic, thus generalizing earlier results for cosine wave functions. We also provide real-life examples and discuss the advantages, for these examples, of considering such non-harmonic waveforms.

AN ADAPTIVE NUMERICAL STRATEGY FOR THE MEDIUM-FREQUENCY ANALYSIS OF HELMHOLTZ'S PROBLEM

The variational theory of complex rays (VTCR) is a wave-based predictive numerical tool for medium-frequency problems. In order to describe the dynamic field variables within the substructures, this approach uses wave shape functions which are exact solutions of the governing differential equation. The discretized parameters are the number of substructures (h) and the number of wavebands (p) which describe the amplitude portraits. Its capability to produce an accurate solution with only a few degrees of freedom and the absence of pollution error make the VTCR a suitable numerical strategy for the analysis of vibration problems in the medium-frequency range. This approach has been developed for structural and acoustic vibration problems. In this paper, an error indicator which characterizes the accuracy of the solution is introduced and is used to define an adaptive version of the VTCR. Numerical illustrations are given.

A droplet entrainment model based on the force balance of an interfacial wave in two-phase annular flow

Droplets are generated at the interface of annular flow due to an interaction between a liquid film and gas core flow. Therefore, knowledge of the interfacial wave structure is essential for making an accurate prediction of the amount of entrained droplets. A new droplet entrainment model was proposed based on the force balance of interfacial waves in vertical annular flow. An analytic wave shape function was developed reflecting the detailed experimental findings, and was used in the development of a new model. The model was validated using the experimental data reported by Hewitt and Pulling at low pressures and by Keeys et al. at high pressures, which had been performed in adiabatic vertical tubes. The root-mean-square error of the prediction of the amount of entrainment was approximately 27% when the model was implemented into COBRA-TF code, which is approximately 23% less than that determined by the Würtz model. The models proposed by Okawa et al. and Stevanovic et al. were also implemented into COBRA-TF and compared with the proposed model.

Elastodynamic infinite elements with united shape functions for soil–structure interaction

In this paper, new formulations of 2D four and five node elastodynamic infinite elements with united shape functions ( EIEUSF) are proposed. Such elements can be treated as family elastodynamic infinite elements, appropriate for dynamic soil–structure interaction (SSI) problems and can be used directly as finite elements. The common characteristic of the proposed infinite elements is the so-called united shape function, which is based on a finite number of wave shape functions. The idea of the united shape function is presented and discussed in details. It is demonstrated that if only one wave function is used (only one wave frequency) these elements are transformed to single-wave elastodynamic infinite elements. The basic difference between the four and five node formulations is the number of nodes for interpolation of the displacement field in the finite direction of the element. The EIEUSF elements are given in decay and mapped form. Fundamental advantage of the proposed infinite elements is that such elements can directly be used in the FEM concept. Once generated the EIEUSF mass and stiffness matrices can be addressed, introduced appropriately in the FEM equation of motion. After the mapping such elements can be recognized and treated, in the FEM technology, as finite elements. The elements are tested for multi-wave soil–structure interaction problems. The results give the guarantee for adequate numerical simulation.

Self-consistent solutions for allowed interconnect current density. I. Implications for technology evolution

We comprehend both interconnect electromigration and Joule heating to study for the first time the self-consistent solutions for the maximum allowed interconnect peak current density j/sub peak/ as a function of wave shape. The maximum allowed temperature and j/sub peak/ solutions monotonically increase as the duty cycle r decreases. Self-consistent solutions indicate that there are diminishing returns in the usefulness of increasing the electromigration capability of the interconnect system. While the greatest increase in allowed j/sub peak/ occurs at de stress (r=1), the increases becomes less as r is reduced below 1, and becomes negligible for r<0.01. Further increases in j/sub peak/ will have to come at some future time from technology options which lower the temperatures at which the interconnects operates.

Complex Zeros of a Triangular Approximation to the Glottal Wave

The acoustic volume-velocity wave produced by the human vocal cords frequently approximates a triangular shape. A theoretical examination of the frequency-domain properties of the triangular wave is consequently of interest in speech analysis. The occurrence of spectral zeros in the glottal wave is of particular importance. When they fall proximate to formants, they can have appreciable influence upon the quality of voiced sounds. The present paper derives equations for the complex frequencies of the zeros. Arbitrary values of wave asymmetry (leg ratio) are treated. In a number of cases, for which the asymmetry can be expressed as a ratio of small whole numbers, exact solutions are obtained. In the less simple cases, solutions are approximated by using a digital computer. Detailed nomograms are presented for the complex-frequency loci of the zeros as a function of waveshape.