Wavelet is rapidly decaying wave
like ossicalation that have zero mean.

Short wave which doesn’t last
forever is called wavelet.  Wavelet have
properties to react to subtle  changes,
discontinuities or break-down points contained in a signal

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Wavelet transform is suitable for
stationary and no stationary signal while Fourier transform is suitable for
stationary signal only.

Unlike the Fourier transform
wavelet transform is suitable for studying the local behavior of the signal
like discontinuity and spikes so it is widely used in crack detection.

 The function   is said to be a wavelet if and only if
its Fourier transform ?(?)

satisfies the wavelet
admissibility condition (Mallat 1998) 

                                               
                                                                  
(3)

 

This condition suggests that the
average value of wavelet must be zero i.e area underneath the curve must be
zero or we can also say that the energy is equally distributed in positive and
negative direction. 

                                                                                                                  
(2)

 And
the 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 domain
or spatial domain , here for crack detection in a beam it is in spatial domain.

 ‘u’ is the translation parameter and ‘s’ is
the scale parameter

Translation means shifting the wavelet
along the time or span , scaling means stretching or compressing the wavelet.

Law scale means more compressed wavelet
it is used for  higher frequency and
better localization,

And high scale represents less compressed
or stretched wavelet it is used for lower frequency and gives less accurate
result with comparison to law scale wavelet. Stretched wavelets helps in
capturing the slowly varying changesin signal while compressed wavelets helps
in capturing the abrupt changes.

This scaled wavelet is translated to the
entire length of the signal and gives desired result(detects the defects and
spikes in the signal ).

Stretched wavelets helps in capturing the
slowly varying changesin signal while compressed wavelets helps in capturing
the abrupt changes.

 

 

 

 

 

 

 

 

Vanishing
moment

    f(x) is said to have k vanishing moment if

      =0

Wavelet with higher number of vanishing
moment gives more accurate result. But there are the limitations of this
approach

If 
a  wavelet have k vanishing moment
it means it will not identify the signal with  polynomial
 for example quardratic signal cant be
detected by wavelet with  3 vanishing
moments because the wavelet  with n
vanishing moment is treated as   derivative of the signal with a smoothing
function (x)
at the scale s.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Discrete
wavelet transforms (DWT)

 

 Discrete wavelet transform is ideal for
denoising 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 integer
multiples as m  
where m=1,2,3,4…

The DWT process is equivalent to compare
a signal with discrete multirate filter banks, the signal is first filtered
with special law pass and high pass filter to yield law pass and high pass sub
bands , the law pass sub bands is called as approximation coefficient
represented as A1,A2,A3 so on and the high pass sub band is called as detailed
coefficient represented as D1,D2,D3 and so on, for the next level of iteration
the law pass sub band is iteratively filtered by same processes to yield
narrower sub bands like D2,D3,D4 and so on, the length of the sub band is half
of the length of the preceding sub band.

 

 the detailed coefficients contains the
necessary information about the irregularities present in the signal that leads
to detection of  irregularities due to
crack .

The selection of level
is depends on the length of the signal if the signal length is N then the wavelet
level 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 so
the value of ‘k’ will be  =8.644, so ‘k’ is 8 here