Wavelet transformWavelet is rapidly decaying wave like oscillation thathave zero mean. Short wave which doesn’t last forever is called wavelet. Wavelethas properties to react to subtle changes, discontinuities or break-down pointscontained in a signal.
Wavelet transform is suitable for stationary and non-stationarysignal. It is also suitable for reviewing the local behavior subtle changes ofthe signal, so it is widely used in crack detection. The function is said to be a wavelet if and only ifits Fourier transform ?(?) fulfills the waveletacceptability condition. (4) This condition suggests that:a) The average value of wavelet must be zero i.
e. area underneaththe curve must be zero or we can also say that the energy is equallydistributed in positive and negative direction. (5) b) The Fourier transform of wavelet function at ?=0 mustbe zero ?(0)=0 (6) The continuous wavelet for a signal f(x) can beexpressed as (8) Where the term = is called as mother wavelet. The variable x can be in the time domain or spatialdomain, here for crack detection in a shaft it is in spatial domain. Thevariable ‘u’ is the translationparameter and’s’ is the scale parameter. Translation means shifting the waveletalong the time or span.
Scaling means stretching or compressing the wavelet. Low scaling means more compressed wavelet, which isused for higher frequency and better localization. High scaling represents lesscompressed or stretched wavelet, which is used for lower frequency. Highscaling gives less accurate results withcomparison to law scaling. Stretched wavelets helps in capturing the slowlyvarying changes in signal while compressed wavelets helps in capturing theabrupt changes. This scaled wavelet is translated to the entire length of thesignal and detects the subtle changes. Afterthe scaling and translation vanishing moment are the another crucial factor which affects the local behavior detectingcapacity of the wavelet. A.
Vanishing moment f(x) is said tohave k vanishing moment if =0 (9) Wavelet with higher number of vanishing moment gives moreaccurate result. But there are the limitations of this approach. If a wavelethave k vanishing moments, it means itwill not identify the signal with polynomial.
For example quadratic signal can’tbe detected by wavelet with 3 vanishing moments. Because the wavelet with n vanishing moments is treated as derivative of the signal with a smoothingfunction () = at the scale s. If the signal has asingularity at a certain point, than the differentiation is not possible atthat point whereas entire signal is differentiated. So the wavelet coefficientshave relatively larger values at that point. B. Discrete wavelet transforms (DWT) Discretewavelet transform is ideal for denoising and compressing the signals andimages.
In DWT, scaling is done in theform of where j=1, 2, 3, 4…, and translation occursat integer multiples as m where m=1, 2, 3, 4….. The DWT processis equivalent to passing a signal with discrete multirate filter banks. Thesignal is first filtered with special law pass and high pass filter to yieldlaw pass and high pass sub bands. The output of law pass sub bands are calledas approximation coefficients represented as Aj, and the high pass sub band arecalled as detailed coefficients represented as Dj. For the next level ofiteration the low pass sub band is iteratively filtered by same processes toyield narrower sub bands like D2, D3, D4 and so on. The length of the sub bandis half of the length of the preceding sub bands.
The Fig.3 shows the flow diagram of discrete wavelet transform up to6 levels. The response of the cracked shaft, as shown in the Fig.2, is theinput signal of the wavelet transform. Wavelet transform up to level 6 isapplied on the signal and its coefficients are plotted as shown in Fig.4. Fig.
4. (a) is the approximation level A1 whichdoesn’t contain any information regarding the crack positions. Detailed level 1to 6 are shown in Fig.
4.(b) – 4(g).The detailed coefficient D1 is shown in theFig 4.(b) in which a larger spikes at crack location 222 is clearly seen. Theother detailed coefficients don’t have clear spikes at crack location.
So forselecting the suitable mother wavelet and suitable signal length, analysis ofthe first detailed level coefficient (D1) is sufficient.Theoriginal signal can be obtained by adding all the detailed coefficients and thelast approximation coefficient.