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.

(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.