## 2 Vibration Signal Data Processing and Feature Enhancement
### Wavelet Decomposition Denoising  
In the CWRU bearing test operation environment, the motor bearing vibration signal is affected by a variety of noise sources, including mechanical noise, electromagnetic  
Figure 3: Sliding signal window panning split signal. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sample Label** & **Fault Diameter (mm)** & **Fault Location** & **Training Set/Test Set** \\ \hline
1 & — & trouble-free & 140/60 \\ \hline
2 & & inner ring & 140/60 \\
3 & 0.1778 & rolling element & 140/60 \\
4 & & outer ring & 140/60 \\ \hline
5 & & inner ring & 140/60 \\
6 & 0.3556 & rolling element & 140/60 \\
7 & & outer ring & 140/60 \\ \hline
8 & & inner ring & 140/60 \\
9 & 0.5334 & rolling element & 140/60 \\
10 & & outer ring & 140/60 \\ \hline
11 & & inner ring & 140/60 \\
12 & 0.7112 & rolling element & 140/60 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: CWRU sample set segmentation. interference, environmental vibration, and so on. These noise signals will be mixed with the signals generated by the bearing to form the final vibration signal. Therefore, the presence of noise signals needs to be taken into account when analyzing and processing the CWRU-bearing dataset. In this paper, wavelet denoising is used to suppress the noise interference to extract the bearing fault characteristics and perform accurate fault diagnosis. Wavelet thresholding denoising is a method of signal denoising using the results of the wavelet transform, which is a technique that converts a signal into the wavelet domain by applying wavelet decomposition. It utilizes the wavelet transform to extract the frequency domain information of the signal and suppresses or eliminates the noise components of the signal by thresholding the wavelet coefficients. \[F(jw)=\int_{-\infty}^{+\infty}f(t)e^{-j\omega t}dt=\int_{-\infty}^{+\infty}f(t )[cos(wt)-jsin(wt)]dt \tag{7}\]  
Equation (6) is the Fourier transform, and the wavelet transform is in principle changing the basis function in the Fourier transform to a wavelet. The wavelet transform formula is as follows:  
\[F(a,\tau)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}f(t)\times\psi(\frac{t- \tau}{a})dt \tag{8}\]  
\(\tau\) is the time shift, a is the scaling factor, and \(\Psi(t)\) is the wavelet basis function. The following are the steps to perform wavelet denoising:  
(1) Perform wavelet decomposition: First, select a suitable wavelet basis function according to the application requirements and signal characteristics. Common wavelet basis functions include Daubechies, Haar, symlets, etc. Second, the signal to be denoised is represented as a discrete sequence, \(x(n)\), \(n=0\), \(1\),..., \(N-1\). Then, at each scale, the selected wavelet basis function is used to perform a convolution operation with the signal to obtain the Approximation Coefficients and Detail Coefficients. In a continuous situation:  
\[c_{j}(k)=\sum_{n}h_{\varphi}(n-2k)c_{j+1}(n) \tag{9}\]  
\[d_{j}(k)=\sum_{n}h_{\varphi}(n-2k)c_{j+1}(n) \tag{10}\]  
\(c_{j}(k)\) denotes the kth coefficient in the \(j\)th level of approximation coefficients, \(d_{j}(k)\) denotes the kth coefficient in the \(j\)th level of detail coefficients, and \(c_{(j+1)}(n)\) denotes the nth coefficient in the \(j\)th + 1 st level of approximation coefficients, and furthermore, the \(h_{\varphi}\) and \(h_{\varphi}\) in the equation represent the low-pass and high-pass filter coefficients of the wavelet function (Mother Wavelet), respectively, which determine the characteristics of wavelet decomposition. Typically, the approximation coefficients represent the low frequency components of the signal at that scale, while the detail coefficients represent the high frequency components of the signal at that scale. This is shown in Figures 4 and 5. (2) Calculate the threshold of the wavelet coefficients: select an appropriate threshold selection method, such as hard threshold or soft threshold. Calculate the threshold of each wavelet coefficient according to the selected thresholding method.