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

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