# III. widely used in crack detection. The function is

III. Wavelet transform

Wavelet is rapidly decaying wave like oscillation that
have zero mean. Short wave which doesn’t last forever is called wavelet. Wavelet
has properties to react to subtle changes, discontinuities or break-down points
contained in a signal. Wavelet transform is suitable for stationary and non-stationary
signal. It is also suitable for reviewing the local behavior subtle changes of
the signal, so it is widely used in crack detection.  The function   is said to be a wavelet if and only if
its Fourier transform ?(?) fulfills the wavelet
acceptability condition.

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(4)

This condition suggests that:

a) 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.

(5)

b) The Fourier transform of wavelet function at ?=0 must
be zero

?(0)=0
(6)

The continuous wavelet for a signal f(x) can be
expressed as

(8)

Where the term

=

is called as mother wavelet.

The variable x can be in the time domain or spatial
domain, here for crack detection in a shaft it is in spatial domain. The
variable ‘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. Low scaling means more compressed wavelet, which is
used for higher frequency and better localization. High scaling represents less
compressed or stretched wavelet, which is used for lower frequency. High
scaling gives less accurate results with
comparison to law scaling. Stretched wavelets helps in capturing the slowly
varying changes in signal while compressed wavelets helps in capturing the
abrupt changes. This scaled wavelet is translated to the entire length of the
signal and detects the subtle changes. After
the scaling and translation vanishing moment are the another crucial factor which affects the local behavior detecting
capacity of the wavelet.

A.      Vanishing moment

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

=0                                          (9)

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 moments, it means it
will not identify the signal with  polynomial. For example quadratic signal can’t
be detected by wavelet with 3 vanishing moments. Because the wavelet with n vanishing moments is treated as  derivative of the signal with a smoothing
function () = at the scale s. If the signal has a
singularity at a certain point, than the differentiation is not possible at
that point whereas entire signal is differentiated. So the wavelet coefficients
have relatively larger values at that point.

B.
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 passing 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 output of law pass sub bands are called
as approximation coefficients represented as Aj, and the high pass sub band are
called as detailed coefficients represented as Dj. For the next level of
iteration the low 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 bands. The Fig.3 shows the flow diagram of discrete wavelet transform up to
6 levels. The response of the cracked shaft, as shown in the Fig.2, is the
input signal of the wavelet transform. Wavelet transform up to level 6 is
applied on the signal and its coefficients are plotted as shown in Fig.4.  Fig.4. (a) is the approximation level A1 which
doesn’t contain any information regarding the crack positions. Detailed level 1
to 6 are shown in Fig. 4.(b) – 4(g).The detailed coefficient D1 is shown in the
Fig 4.(b) in which a larger spikes at crack location 222 is clearly seen. The
other detailed coefficients don’t have clear spikes at crack location. So for
selecting the suitable mother wavelet and suitable signal length, analysis of
the first detailed level coefficient (D1) is sufficient.

The
original signal can be obtained by adding all the detailed coefficients and the
last approximation coefficient.