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Hardcover ISBN: Imprint: Academic Press. Published Date: 21st September Page Count: View all volumes in this series: Wavelet Analysis and Its Applications. For regional delivery times, please check When will I receive my book? Sorry, this product is currently out of stock.
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Read the latest chapters of Wavelet Analysis and Its Applications at goystofetel.ga, Elsevier's leading platform of peer-reviewed scholarly literature. Time-Frequency/Time-Scale Analysis - 1st Edition - ISBN: , View all volumes in this series: Wavelet Analysis and Its Applications.
Detailed coverage of both linear and quadratic solutions Various techniques for both random and deterministic signals. Powered by. You are connected as. Connect with:. Many books and papers have been written that explain WT of signals and can be read for further understanding of the basics of wavelet theory.
The concept of wavelets in its present theoretical form was first proposed by J. Grossmann, a theoretical physicist, in France. They provided a way of thinking for wavelets based on physical intuition.
On the Quantum Correction for Thermodynamic Equilibrium. Journal of Sound and Vibration 1—2 : — Faulted EPS signals are associated with fast electromagnetic transients and are typically nonperiodic and with high-frequency oscillations. However, it should always be remembered that the unit of frequency for discrete time signals is radians. All Journals. Both parameters vary in terms of time and frequency, respectively, in signal analysis using WT.
In other words, the transform of a signal does not change the information content presented in the signal [ 1 ]. Thus, in the first part, this chapter presents an overview of the main characteristic of wavelet transform for the transient signal analysis and the application on electric power system. The property of multiresolution in time and frequency provided by wavelets allows accurate time location of transient components while simultaneously retaining information about the fundamental frequency and its low-order harmonics.
This property of the wavelet transform facilitates the detection of physically relevant features in transient signal to characterize the source of the transient or the state of the postdisturbance system. Initially, we will discuss the performance, advantages, and limitations of the WT in electric power system application, where the basic wavelet theory is presented. Additionally, the main publications carried out in this field will be analyzed and classified by the next areas: power system protection, power quality disturbances, power system transient, partial discharge, load forecasting, faults detection, and power system measurement.
Finally, a comprehensive analysis related to the advantages and disadvantages of the WT in relation to other tools is performed. The wavelet transform theory is based on analysis of signal using varying scales in the time domain and frequency.
Formalization was carried out in the s, based on the generalization of familiar concepts. The wavelet term was introduced by French geophysicist Jean Morlet. The seismic data analyzed by Morlet exhibit frequency component that changed rapidly over time, for which the Fourier Transform FT is not appropriate as an analysis tool.
Thus, with the help of theoretical physicist Croatian Alex Grossmann, Morlet introduced a new transform which allows the location of high-frequency events with a better temporal resolution [ 2 ]. Faulted EPS signals are associated with fast electromagnetic transients and are typically nonperiodic and with high-frequency oscillations. This characteristic presents a problem for traditional Fourier analysis because it assumes a periodic signal and a wide-band signal that require more dense sampling and longer time periods to maintain good resolution in the low frequencies [ 3 ].
The WT is a powerful tool in the analysis of transient phenomena in power system.
It has the ability to extract information from the transient signals simultaneously in both time and frequency domains and has replaced the Fourier analysis in many applications [ 4 ]. The functions based term refers to a complete set of functions that, when combined on the sum with specific weight can be used to then construct a certain sign [ 5 ]. The main characteristic of the WT is that it uses a variable window to scan the frequency spectrum, increasing the temporal resolution of the analysis.
The wavelets are represented by:. In Eq. Thus, it is evident that WT has a zero rating property that increases the degrees of freedom, allowing the introduction of the dilation parameter of the window [ 8 ]. The continuous wavelet transform CWT of the continuous signal x t is defined as:. The WT coefficient is an expansion and a particular shift represents how well the original signal x t corresponds to the translated and dilated mother wavelet. Thus, the coefficient group of CWT a,b associated with a particular signal is the wavelet representation of the original signal x t in relation to the mother wavelet [ 9 ].
The redundancy of information and the enormous computational effort to calculate all possible translations and scales of CWT restricts its use. An alternative to this analysis is the discretization of the scale and translation factors, leading to the DWT. There are several ways to introduce the concept of DWT, the main are the decomposition bands and the decomposition pyramid or Multi-Resolution Analysis -MRA , developed in the late s [ 10 ]. The DWT of the continuous signal x t is given by:. The problems of temporal resolution and frequency found in the analysis of signals with the STFT best resolution in time at the expense of a lower resolution in frequency and vice-versa can be reduced through a multi-resolution analysis MRA provided by WT.
Both parameters vary in terms of time and frequency, respectively, in signal analysis using WT. In the STFT, a higher temporal resolution could be achieved at the expense of frequency resolution. Intuitively, when the analysis is done from the point of view of filters series, the temporal resolution should increase increasing the center frequency of the filters bank. The result is geometric scaling, i. The CWT follows exactly these concepts and adds the simplification of the scale, where all the impulse responses of the filter bank are defined as dilated versions of a mother wavelet [ 10 ].
The CWT is a correlation between a wavelet at different scales and the signal with the scale or the frequency being used as a measure of similarity. The CWT is computed by changing the scale of the analysis window, shifting the window in time, multiplying by the signal, and integrating over all times. In the discrete case, filters of different cut-off frequencies are used to analyze the signal at different scales. The signal is passed through a series of high-pass filters to analyze the high frequencies, and it is passed through a series of low-pass filters to analyze the low frequencies.
The Mallat algorithm consists of series of high-pass and the low-pass filters that decompose the original signal x [ n ] into approximation a n and detail d n coefficient, each one corresponding to a frequency bandwidth. The resolution of the signal, which is a measure of the amount of detail information in the signal, is changed by the filtering operations, and the scale is changed by up-sampling and down-sampling sub-sampling operations.
Sub-sampling a signal corresponds to reducing the sampling rate or removing some of the samples of the signal. For other hand, up-sampling a signal corresponds to increasing the sampling rate of a signal by adding new samples to the signal. The procedure starts with passing this signal x [ n ] through a half band digital low-pass filter with impulse response h [ n ]. Filtering a signal corresponds to the mathematical operation of convolution of the signal with the impulse response of the filter.
The convolution operation in discrete time is defined as follows [ 2 ]:. A half band low-pass filter removes all frequencies that are above half of the highest frequency in the signal. However, it should always be remembered that the unit of frequency for discrete time signals is radians. Simply discarding every other sample will subsample the signal by two, and the signal will then have half the number of points.
The scale of the signal is now doubled. Note that the low-pass filtering removes the high frequency information but leaves the scale unchanged. Only the sub-sampling process changes the scale. Resolution, on the other hand, is related to the amount of information in the signal, and therefore, it is affected by the filtering operations. Half band low-pass filtering removes half of the frequencies, which can be interpreted as losing half of the information. Therefore, the resolution is halved after the filtering operation.
Note, however, the sub-sampling operation after filtering does not affect the resolution, since removing half of the spectral components from the signal makes half the number of samples redundant anyway. Half of the samples can be discarded without any loss of information. This procedure can mathematically be expressed as [ 2 ]:. The decomposition of the signal into different frequency bands is simply obtained by successive high-pass and low-pass filtering of the time domain signal.
The original signal x [ n ] is first passed through a half band high-pass filter g [ n ] and a low-pass filter h [ n ]. The signal can therefore be sub-sampled by 2, simply by discarding every other sample. This constitutes one level of decomposition and can mathematically be expressed as follows [ 2 ]:. In [ 12 ], an application of WT and its advantages compared to Fourier transform is presented. One of the main advantages of wavelets is that they offer a simultaneous localization in time and frequency domain.
The second main advantage of wavelets is that, using fast wavelet transform, it is computationally very fast. Wavelets have the great advantage of being able to separate the fine details in a signal. Very small wavelets can be used to isolate very fine details in a signal, while very large wavelets can identify coarse details. A wavelet transform can be used to decompose a signal into component wavelets. In wavelet theory, it is often possible to obtain a good approximation of the given function f by using only a few coefficients, which is a great achievement when compared to Fourier transform.
Wavelet theory is capable of revealing aspects of data that other signal analysis techniques miss like trends, breakdown points, and discontinuities in higher derivatives and self-similarity. It can often compress or de-noise a signal without appreciable degradation [ 12 ]. The Fourier transform shows up in a remarkable number of areas outside classic signal processing. Even taking this into account, we think that it is safe to say that the mathematics of wavelets is much larger than that of the Fourier transform.
In fact, the mathematics of wavelets encompasses the Fourier transform. The size of wavelet theory is matched by the size of the application area. Initial wavelet applications involved signal processing and filtering. However, wavelets have been applied in many other areas including nonlinear regression and compression. An offshoot of wavelet compression allows the amount of determinism in a time series to be estimated [ 12 ]. These authors also present a literature review on the application of WT in power electrical systems.
By means of the bibliographic review, it is possible to highlight certain topics of interest for researchers: Power quality.