# Review of Recent Advances in the Application of the Wavelet Transform to Diagnose Cracked Rotors
## 2 The Wavelet Transform
### Continuous Wavelet Transform  
The same way Fourier Transform (FT) obtains correlation coefficients between the analyzed signal and a sinusoidal one, WT obtains correlation coefficients between the signal and an orthonormal function, called the wavelet function, depending on the scale and position of the wavelet function. There are a lot of different types of wavelet functions, such as those detailed by [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON].  
The Continuous Wavelet Transform (CWT) allows analysis of a signal through the correlation coefficients of that signals instead of using the whole signal information. The mathematical formula for determining CWT is shown in Equation (1):  
\[CWT(c,b;\psi)=w\int_{-\infty}^{\infty}x(t)\psi^{*}(\frac{t-b}{c})dt \tag{1}\]  
where \(x(t)\) is the signal in the time domain, \(\psi\) is the wavelet function, and \(w\) is a weighting function. The parameter \(c\) is related to the scale, and \(b\) is related to the position of the wavelet function. \(CWT(c,b;\psi)\) represents the resulting coefficients, as a function of \(c\), \(b\) and the wavelet function \(\psi\).  
Figure 1 shows a signal in the time domain and the corresponding CWT.  
CWT is one of the best tools available to detect singularities of a signal. Singularity detection is carried out with the local maxima lines [25].  
Regarding crack detection, [PERSON] _et al._[41] affirmed that the presence of sub-critical (1/3 and 1/2 of the critical speed) components in the CWT may be indicative of cracking, although this may  
Figure 1: Signal in the time domain and its CWT.  
not always be reliable due to the high sensitivity to noise. They proposed an alternative technique, involving the calculation of the phase angle between the two signals (cracked and un-cracked) of two transverse vibration. The phase angle can be calculated with the Cross Wavelet Transform (XWT), called as they proposed.  
The sub-critical \(1/3\) and \(1/2\) components calculated with CWT have also been used as indicators of the of a crack [42]. The amplitude values of sub-critical and critical peaks in the CWT are used to feed an Artificial Neural Network (ANN) [43], allowing crack diagnosis, and determination of the position and depth of the crack.  
CWT has also been applied to diagnose notched rotors (_i.e._, rotors with a transverse open crack) [44], where it has been demonstrated that both CWT and changes on the \(2\times\) harmonic can be used as robust indicators.  
[PERSON] _et al._[45] used the CWT coefficients as the input for an ANN. They studied the ability of the system to detect combined faults of unbalance and shaft crack. This method was tested successfully with a hit rate of 99.9%.  
CWT has been used in engineering applications for fault detection of rotating machinery in the form of a scalogram. The scalogram is defined as the square of the modulus of the CWT. However, the use of CWT in current applications, to diagnose machinery faults, is still relatively rare. This is due to the fact that the visual interpretation of wavelet results is often difficult. Efforts have been made to extract the best features analyzing the residual wavelet scalogram [46].