Wavelet zero ?(0)=0 (3) The continuous wavelet for

Wavelet is rapidly decaying wavelike ossicalation that have zero mean. Short wave which doesn’t lastforever is called wavelet.  Wavelet haveproperties to react to subtle  changes,discontinuities or break-down points contained in a signalWavelet transform is suitable forstationary and no stationary signal while Fourier transform is suitable forstationary signal only.Unlike the Fourier transformwavelet transform is suitable for studying the local behavior of the signallike discontinuity and spikes so it is widely used in crack detection. The function   is said to be a wavelet if and only ifits Fourier transform ?(?)satisfies the waveletadmissibility condition (Mallat 1998)                                                                                                                   (3)  This condition suggests that theaverage value of wavelet must be zero i.e area underneath the curve must bezero or we can also say that the energy is equally distributed in positive andnegative direction.

                                                                                                                    (2) Andthe Fourier transform of wavelet function at ?=0 must be zero                                                    ?(0)=0                                                                                           (3)The continuous wavelet for a signal f(x) with mother wavelet   =Can be expressed as    The variable x can be in the time domainor spatial domain , here for crack detection in a beam it is in spatial domain. ‘u’ is the translation parameter and ‘s’ isthe scale parameter Translation means shifting the waveletalong the time or span , scaling means stretching or compressing the wavelet.Law scale means more compressed waveletit is used for  higher frequency andbetter localization,And high scale represents less compressedor stretched wavelet it is used for lower frequency and gives less accurateresult with comparison to law scale wavelet. Stretched wavelets helps incapturing the slowly varying changesin signal while compressed wavelets helpsin capturing the abrupt changes.This scaled wavelet is translated to theentire length of the signal and gives desired result(detects the defects andspikes in the signal ).Stretched wavelets helps in capturing theslowly varying changesin signal while compressed wavelets helps in capturingthe abrupt changes.        Vanishingmoment     f(x) is said to have k vanishing moment if       =0Wavelet with higher number of vanishingmoment gives more accurate result. But there are the limitations of thisapproach If a  wavelet have k vanishing momentit means it will not identify the signal with  polynomial for example quardratic signal cant bedetected by wavelet with  3 vanishingmoments because the wavelet  with nvanishing moment is treated as   derivative of the signal with a smoothingfunction (x)at the scale s.

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                             Discretewavelet transforms (DWT)  Discrete wavelet transform is ideal fordenoising and compressing the signals and images.  In DWT Scaling is done in the form of  where j=1,2,3,4…And translation occurs at integermultiples as m  where m=1,2,3,4… The DWT process is equivalent to comparea signal with discrete multirate filter banks, the signal is first filteredwith special law pass and high pass filter to yield law pass and high pass subbands , the law pass sub bands is called as approximation coefficientrepresented as A1,A2,A3 so on and the high pass sub band is called as detailedcoefficient represented as D1,D2,D3 and so on, for the next level of iterationthe law pass sub band is iteratively filtered by same processes to yieldnarrower sub bands like D2,D3,D4 and so on, the length of the sub band is halfof the length of the preceding sub band.  the detailed coefficients contains thenecessary information about the irregularities present in the signal that leadsto detection of  irregularities due tocrack . The selection of levelis depends on the length of the signal if the signal length is N then the waveletlevel should be ‘k’ such that  , if ‘k’ is in decimal  then only integer value of ‘k’ is taken. Here the length of the signal is 400 sothe value of ‘k’ will be  =8.644, so ‘k’ is 8 here

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