## 4 Wavelet Transform
### Wavelet Transform Method
#### Discrete Wavelet Transform  
The DWT was introduced to simplify the computational complexity of the CWT. To address the redundancy and computational complexity inherent in the CWT, the DWT was developed by discretizing both the scale parameter \(a\) and the translation parameter \(b\), thereby achieving \(a\) balance between theoretical completeness and practical implementation in the following Equation (25):  
\[\psi_{a,b}=\frac{1}{\sqrt{2^{m}}}\psi\bigg{(}\frac{t-n}{2^{m}}\bigg{)}, \tag{25}\]  
To perform approximate analysis, it is essential to select an appropriate level \(m\) for either a detailed or approximate further analysis, as represented by the DWT in Equation (26). \[DWT_{f}(m,n)=\frac{1}{\sqrt{2^{m}}}{\sum_{k}}f(k)\psi^{*}\bigg{(}\frac{k-n}{2 ^{m}}\bigg{)}, \tag{26}\]  
Here, \(DWT_{f}(m,n)\) represents the discrete wavelet transform coefficients of the signal \(f(t)\), and \(k\) is the operation index. It is noteworthy that while the DWT is discrete in nature,it is essentially a transformation of a continuous signal, with discretization limited to the scale parameter \(a\) and translation parameter \(b\). The DWT can be considered a decomposition method that analyzes the original signal at multiple resolutions. By employing high-pass and low-pass filters, the DWT decomposes the original signal into high-frequency and low-frequency components. The signal is initially passed through high-pass and low-pass filters, followed by down-sampling by a factor of two to obtain the high-frequency and low-frequency coefficients, respectively. By applying Equation (27), which relates the sampling frequency \(f_{e}\) and fundamental frequency \(f_{s}\), the number of levels \(n_{ls}\) in the discrete wavelet transform can be obtained. \[n_{ls} =\text{int}(\frac{\log\frac{f_{e}}{f_{s}}}{\log 2}), \tag{27}\]  
Wavelet functions, essentially bandpass filters, halve their bandwidth at each scale. To cover the entire spectrum, an infinite number of levels would be required. However, in practical applications, the number of analyzable levels is limited by the sampling quantity. Therefore, the scaling function (27) is employed to filter out the lowest level of the transform, ensuring the effective coverage of the entire spectrum. In addition to performing the DWT, signal reconstruction is also involved. This process includes passing the signal through a low-pass filter and a high-pass filter to obtain the high-frequency and low-frequency components of the signal. The reconstruction of the original signal, known as the IDWT, involves applying the inverse transform filtering operation to recover the original signal. The reconstruction process, which is the inverse of the aforementioned decomposition process, involves using the obtained details from each level to reconstruct the original sequence. This is achieved by up-sampling the sequences and passing them through a series of \(n_{ls}\) two-band perfect reconstruction quadrature mirror filter banks (QMF). This process is known as the inverse discrete wavelet transform (IDWT). Since the motor voltage signal, after calculating the details of each level using DWT, cannot be directly used for fault feature analysis, IDWT is required to perform the inverse transform. Through up-sampling, the reconstructed waveforms at different frequency ranges are aligned to a consistent length, facilitating subsequent computations. Therefore, the IDWT comprises high-pass and low-pass filters and plays a crucial role in signal reconstruction. The hierarchical structure of the DWT process is illustrated in Figure 3.  
The signal source, \(X(n)\), can be decomposed into a representation using wavelet functions as an orthonormal basis, as shown in Equation (28). \[X(t)=\sum_{k}cA_{i}(k)\phi_{j-i,k}+\sum_{k}cD_{i}(k)w_{j-k,k}+\ldots+\sum_{k}cD_ {1}(k)w_{j-k,k}, \tag{28}\]  
where the scaling function and wavelet function are represented by the following Equations (29) and (30), respectively. Figure 3: Discrete wavelet transform steps.