## 3 Signal Preprocessing Techniques  
The vibration signals of the motors are non-stationary and vary over time. Neither frequency-domain nor time-domain techniques alone can provide sufficient information for the effective diagnosis of these signals. The traditional frequency analysis method, FFT, provides frequency information, expressed as Equation (1) [32]. However, there are multiple frequency components in the physical signal of stator faults, making it difficult to distinguish the relationship between these frequencies and bearing fault frequencies. Based on FFT, the method requires a large amount of data and high computational cost to increase the sample rate when the samples are the same [33]. However, the spectrogram obtained after time-frequency analysis can be used to identify the characteristics of stationary or non-stationary signals. Time-frequency analysis captures how frequency components vary over time. The time-frequency characteristics will also be significantly different in the time-frequency domain under both healthy and faulty conditions [34]. Therefore, the spectrogram is a useful technique for effectively analyzing the physical signals generated by mechanical components with various frequency components [35]. A common time-frequency transform technique is continuous wavelet transform (CWT). The CWT is a time-frequency transform method. By scaling and translating the mother wavelet function, CWT can analyze signals at different times and scales. Because the scaling factor allows for the dynamic adjustment of time and frequency resolutions, CWT exhibits multi-resolution characteristics, enabling it to achieve high resolution in both the time and frequency  
Figure 1: Types of faults.  
(scale) domains, simultaneously. Therefore, CWT is particularly suitable for analyzing non-stationary signals.  
\[X[h]=\sum_{k=0}^{K-1}x[k]e^{-\frac{j\frac{2\pi ik}{K}}{K}} \tag{1}\]  
where real discrete signals \(x[k]\), having a finite length \(K\), are derived from the sampling of continuous signals and \(j\) is the imaginary unit.  
Multi-resolution capability in time-frequency distribution plots is crucial for accurately diagnosing motor faults from vibration signals. The CWT has been developed into a time-frequency analysis method for stationary and non-stationary signals. Since the translation and scaling of the mother wavelet function can provide fine resolution in time and frequency, it can fully demonstrate the energy distribution from a low frequency to a high frequency.  
The CWT can transform the time sequence signals to the time-frequency domain. The CWT process decomposes the time sequence signals by scaling and translation, expressed as Equation (2) [36]. The signals will be decomposed into linear combinations using the mother wavelet function (different scaling and translation) as the basis function. In addition, as the basis function has \(\tau\) and \(s\) parameters, the CWT can obtain the relationship between time and frequency, and the signal frequency range in the period can be observed; the transformation process is shown in Figure 2.  
\[W(\tau,s)=\frac{1}{\sqrt{s}}{\int_{-\infty}^{\infty}x(t)\psi\left(\frac{t-\tau }{s}\right)t} \tag{2}\]  
where \(\tau\) is the translation parameter, time translation is available, so the location of time can be provided, \(s\) is the scale parameter, representing the mother wavelet scaling factor, \(x(t)\) is the time sequence signal, and \(\psi(t)\) is the mother wavelet function.