# Review of Recent Advances in the Application of the Wavelet Transform to Diagnose Cracked Rotors
## 3 Wavelet Transform Parameters Selection  
One of the main parameters to take into consideration when using wavelet theory is the selection of the frequency resolution in each zone of the signal. This is determined by the scale range in CWT, and by the decomposition level in DWT. Figure 4: WPT decomposition process until level 3 of decomposition, where each decomposition is performed based on DWT (see Figure 2). When using CWT, there are different resolution in frequency and in time at the different zones of the signal, therefore the scale range must be adjusted to get the desired frequency range at a local zone of the signal. The scale range determines the frequency range. The relationship between the scale, \(s\), and the frequency, \(f\), when applying CWT is shown in Equation (3). \[f=\frac{f_{\psi}\cdot f_{s}}{s} \tag{3}\]  
where \(f_{\psi}\) is the central frequency of the wavelet function in Hz, and \(f_{s}\) the sampling frequency. The analysis of the standard deviation of the wavelet scales energy is one of the most important features in distinguishing impulsive and oscillatory transient disturbance, as performed in [61] were a power system transient analysis was performed. On the other hand, when using DWT, every time the signal is decomposed, the frequency range is divided by two, as demonstrated in Figure 2. When the MRA is applied, not all of the packets are decomposed, due to this the frequency ranges of the different areas of the signal are different. As opposed to MRA, when using WPT, all the packets obtained at each level have the same frequency resolution, until the desired one is reached. Considering the application of WPT using a decomposition level, \(k\), and the global frequency resolution, \(f_{s}/2\), (half of the sampling frequency \(f_{s}\) in accordance with the Nyquist rule), the frequency resolution of the whole signal is divided into equal parts between the number of packets obtained \(2^{k}\). Thus, the frequency resolution, \(f_{r}\), of each packet is given by Equation (4) [36]:  
\[f_{r}=\frac{f_{s}/2}{2^{k}} \tag{4}\]  
The value of \(f_{r}\) must achieve a compromise between covering the desired range whilst also being small enough to avoid being influenced by other frequencies that may adulterate the results. In most cases, the decomposition level selected is optimal for one purpose and has been reached after trying several other options [62]. Since all WPT packets have the same frequency resolution, the automatic selection of the optimal scale range or decomposition level becomes easier. An automatic method has been detailed [63] for the selection of features involving sets that produce accurate results in classification for systems. The method utilizes wavelet decomposition and texture statistics. The use of intelligent classification systems has also been used for feature selection (either single-level basis selection, or multiple-level basis selection). Classification results of SVM have also been used at different Signal to Noise Ratios for feature selection [64]. Based on this work, a method to automatically select the optimal decomposition level was proposed in [58]. The method is based on the diagnosis results provided by an ANN. The method proposes an automatic evaluation to diagnose cracks in shafts, trying decomposition levels from two to nine. Another critical parameter to be selected when applying wavelets is the wavelet function. In the same way it was common to determine the decomposition level of the range scale selection, it is also common to experiment with several possibilities to try to get the optimum wavelet function for the desired purpose. In terms of data compression, several methods are used, as Compression Ratio and Peak Signal to Noise Ratio, this is demonstrated in [65]. In most cases, the selection of the decomposition level and the wavelet function are closely related. For example, when the DWT is applied to remove noise, there are several methods proposed, such as the Signal to Noise Ratio, MAX, and EXP methods [66].