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Application of Integral Transforms in Flood Studies
Published in Saeid Eslamian, Faezeh Eslamian, Flood Handbook, 2022
Vahid Nourani, Mehran Dadashzadeh, Saeid Eslamian
The continuous wavelet transform (CWT) is similar to the Fourier transform in the sense that it is based on a single function ψ and that this function is scaled. But unlike the Fourier transform, we also shift the function, thus generating a two-parameter family of functions ψa,b. It is convenient to define ψa,b as follows: ψa,bx=a−12ψx−ba.
Advanced Harmonic Analysis for Power Systems
Published in Felix Alberto Farret, Marcelo Godoy Simões, Danilo Iglesias Brandão, Electronic Instrumentation for Distributed Generation and Power Processes, 2017
Felix Alberto Farret, Marcelo Godoy Simões, Danilo Iglesias Brandão
The wavelet theory applies to many subjects. All wavelet transforms may be considered forms of time-frequency representation for continuous signals (analog), being so related to harmonic analysis. Practically, almost all discrete wavelet transforms use banks of filters in time. These filter banks are called scaling coefficients in the wavelet nomenclature and may contain filters either with a finite impulse response (FIR) or infinite impulse response (IIR). The wavelets forming a continuous wavelet transform (CWT) are subject to the principle of Fourier analysis of the uncertainty regarding the theory of sampling; i.e., given a signal with an event on it, one cannot simultaneously set a response scale in time and frequency for this event. The time response scale of all uncertainty, with regards to this particular event, has lower bandwidth. Thus, the scalogram of a continuous wavelet transform of this signal, such as an event, marks an entire region in the time-scale plan rather than of a single point. Similarly, the basis of the discrete wavelet can be considered in the context of other forms of the uncertainty principle.
Principles and Applications of Plasma Actuators
Published in Ranjan Vepa, Electric Aircraft Dynamics, 2020
Wavelet transforms and wavelet packet decompositions [35] have recently gained popularity for the analysis of fluid-flow signals. Wavelet packet decomposition is particularly ideally suited for de-noising [36] a measured fluid-flow signal. Over the past ten years, wavelet transform has been used as a powerful tool for image data compression, noise reduction and feature extraction of a signal. Wavelets provide efficient localization in both time and frequency (or scale). To analyze any finite energy signal, the continuous wavelet transform (CWT) provides a decomposition of the signal as a combination of a set of basis function, obtained by means of dilation and translation of a single prototype wavelet function called a mother wavelet. There are over 300 different types of mother wavelets including Haar, Daubechies (db), Symlet, Coiflet, Gaussian, Morlet, complex Morlet, Mexican hat, bio-orthogonal, reverse bio-orthogonal, Meyer, discrete approximation of Meyer, complex Gaussian, Shannon and frequency B-spline families. The continuous wavelet transform maps a signal of one independent variable into another function of two independent variables representing a scaling and translation of the independent variable. The scale factor and/or the translation parameter can both be discretized. The discretization process leads to an orthonormal basis for signal representations. The decomposition process can be iterated, with successive approximations being decomposed in turn, so that one signal is broken down into many lower resolution components. For discrete-time signals, the dyadic discrete wavelet transform (DWT) is equivalent, according to Mallat’s algorithm [37] to an octave filter bank, and can be implemented as a cascade of identical cells (low-pass and high-pass finite impulse response filters). These filters split the signal’s bandwidth to half. Using downsamplers after each filter, the redundancy of the signal representation can be removed. This is called the wavelet decomposition tree and is the basis for wavelet decomposition.
Wavelet-based operating deflection shapes for locating scour-related stiffness losses in multi-span bridges
Published in Structure and Infrastructure Engineering, 2023
Eugene J. OBrien, Daniel P. McCrum, Muhammad Arslan Khan, Luke J. Prendergast
The main advantage of the continuous wavelet transform (CWT) is its ability to provide dynamic information simultaneously in the frequency and time domains with adaptive windows. Wavelet transforms contain local singularity information, which enables the detection of damage location and severity (Zhu & Law, 2006). ODS contain the contributions from all operating modes at any frequency (Schwarz & Richardson, 1999) and can be analysed using a wavelet transform. In this way, ODS can be used to locate damage in the affected structure. However, it should be noted that in the process of obtaining wavelet transforms, the so-called boundary distortion phenomenon (Asnaashari & Sinha, 2014) occurs due to the finite length of the signal near boundaries. This phenomenon influences the ability of wavelets to detect singularities, or damage, occurring near the bridge boundaries.
Extraction characteristics of pressure signals of oil pulsating flow with different particle concentration based on wavelet analysis
Published in Petroleum Science and Technology, 2022
Since the particle quantity of particle pollutants mixed in oil is a random distribution problem, the dynamic flow parameters in the time domain and frequency domain show a trend of drastic changes. The wavelet analysis can make a group of data that have good local properties in the time domain and frequency domain, the mutation information in a group of time series can be detected, and its main change periods can be analyzed; meanwhile, the influence of these periods in different periods can be obtained. Wavelet analysis mainly uses the continuous wavelet transform (CWT) to detect transient signals, and provides powerful multi-resolution in time-frequency analysis (Belaid, Miloudi, and Bournine 2021). CWT has a set of basic functions that decompose a signal into different localization levels in time and scale (Zhou et al., 2020). These operations generate wavelet coefficients through the expansion and translation of mother wavelets. Lu et al. (2017) analyzed the multi-scale decomposition of the velocity data of Particle Image Velocimetry (PIV) using the wavelet analysis method. They investigated the influence of surfactants on the turbulence characteristics of the multi-scale channels.
Applicability analysis of IR thermography and discrete wavelet transform for technical conditions assessment of bridge elements
Published in Quantitative InfraRed Thermography Journal, 2019
Leszek Różański, Krzysztof Ziopaja
Theoretical background and practical applications of Wavelet Transform were published in the early 1990s by Ingrid Daubechies, C.K. Chui, Y. Mayer and D.E. Newland. Wavelet Transform (WT) is a mathematical tool, which has been successfully used in active IR thermography [19] for the analysis of thermographic data. The main advantage of WT is its usefulness in damage identification problem solving, because it discovers and locates information in time or space on local disturbances and discontinuities in the signal that may be caused by hidden defects. Continuous wavelet transform (CWT) due to the time-consuming calculation, a very large (infinite) number of factors and because of the small number of wavelet in the analytical form, has a limited field of application in practical engineering tasks. DWT does not have the above drawbacks. In many engineering cases the analyzed one- and two-dimensional signal has a discrete form. The algorithms used for calculating the signal decomposition and reconstruction are simple and effective. The wavelets family used to DWT is large and allows to more easily choose appropriate wavelet for the specific problem. In this article, the temperature distribution (real or numeric) is treated as a set of temperature values from the area of, e.g. N = 64x64 = 4096. It means that the level of decomposition of the signal is equal to 6, which corresponds to the number of details which contain the various features of the signal. In 1989 Stephane Mallat presented the algorithm of multiresolution decomposition of two-dimensional signal F(x,y) ∊L2 in the wavelet representation: