# Hybrid Wavelet-CNN Fault Diagnosis Method for Ships' Power Systems
## 4 Research Methodology  
### Discrete Wavelet Analysis  
There are several types of wavelet transformations, each with its own set of applications, but the two main ones are continuous wavelet transform (CWT) and discrete wavelet \begin{table}
\begin{tabular}{c c c} \hline
**Duration (s)** & **Load Torque (Nm)** & **Rotational Speed (rpm)** \\ \hline
0–1 & 0 & 1499 \\
1–2 & 26.72 & 1434 \\
2–3 & 13.36 & 1468 \\
3–4 & 6.68 & 1484 \\
4–5 & 0 & 1499 \\ \hline \end{tabular}
\end{table} 
Table 2: Different load conditions of step load torque. Figure 3: Time series of 3–phase stator currents during faulty operation. Figure 2: Time series of 3–phase stator currents during healthy operation. transform (DWT), as highlighted in [9; 12; 24]. The CWT is mathematically expressed with the following equation [24]:  
\[W(a,\ b)=\frac{1}{\sqrt{a}}\int\limits_{-\infty}^{+\infty}f(x)\varphi\left(\frac {x-b}{a}\right)dx \tag{1}\]  
where \(\phi\) is the mother wavelet, \(a\) is the scale factor, and \(b\) is the translation factor. The values of the factors are continuous, which means that there is an infinite number of wavelets. The main difference of DWT is that discrete values are used for the scale and translation factors. DWT is mathematically expressed with the following equation [24]:  
\[W(a,\ b)=2^{-(\frac{a}{2})}\sum\limits_{t=0}^{T-1}f(t)\varphi\left(\frac{t-b2 ^{a}}{2^{a}}\right) \tag{2}\]  
DWT can be utilized as a filter bank, meaning that the signal is decomposed into several frequency sub-bands ([PERSON] multidecomposition theory). According to the authors of [9], the DWT is implemented by convoluting the signal with sub-band low- and high-pass filters. The output of the filters returns the DWT coefficients. Decomposition theory is applied to many levels. At the first level, the signal is split into a high- and a low-frequency part to analyze it at its maximal frequency. At the second level, the previous low-frequency part is again split into high- and low-frequency parts in order to analyze it at half of its maximal frequency. This process is repeated until the maximal decomposition level, ending up with a set of approximation and detail coefficients from each level. The coefficients can be used to reconstruct the original signal. In some cases, it is desirable to exclude detail coefficients from various levels to reconstruct the original signal without the high-frequency behavior (denoising). Harmonic frequencies are generated during faulty operation in the current; thus, in our case, it was desirable to reconstruct the original signal by keeping the detail coefficients from the appropriate levels at which faulty operation was indicated if it existed. Additionally, the selection of the appropriate mother wavelet for signal representation is crucial. There are several DWT families and they are categorized on the basis of the features of the produced basis functions. Examples of different families are Haar, the discrete approximation of Meyer, Symlet, and Daubechies. There were cases in our simulation where the fault was indistinguishable in the time series of the currents (Figure 4). DWT was utilized to solve this problem since it is ideal for the analysis of nonstationary signals [5]. As mentioned above, the selection of an appropriate type of Wavelet is significant. In addition, the level of decomposition is important for the efficient implementation of the DWT. Reconstructed signals from various detail coefficients are depicted in Figure 5 for the case of an indistinguishable fault in the time series. On the basis of the descriptive analysis in Figure 5 and after experimenting with various types of DWT, we concluded that the discrete approximation of Meyer (Dmey) was a good option. Dmey is ideal for the computation of fast discrete wavelet transforms and it does not exhibit many disadvantages compared to other DWT types [25].