## 2 An Overview of Wavelet Analysis  
When interacting with a multi-component signal with a frequency that changes over time, it is necessary to represent that signal in the time-frequency domain [29]. The main advantage of time-frequency analysis compared to other methods is its ability to represent a vibration signal simultaneously in the time and frequency domains (called the time-frequency domain), providing crucial information about the frequency structure of the signal and allowing the observation of the temporal variations of the frequency [30]. The signal is transformed by an integral transform similar to the Fourier transform, known as a time-frequency represented (TFR). Time-frequency transformations are classified into linear and nonlinear. However, time-frequency non-linear transformations have drawbacks regarding algorithmic complexity, resulting in slow computation due to many operations and difficulties in interpreting the results. As a result, they have rarely been used to analyze mechanical vibration.  
The wavelet transform is categorized as a linear transformation. It is divided into two groups, continuous wavelet transform (CWT) and discrete wavelet transform (DWT), based on the computation method. Recent advancements in CWT and DWT have improved signal resolution to obtain wavelet packet transform (WPT), second-generation wavelet transform, and wavelet-based synchrosqueezing transform (WSST). The methods of signal analysis based on CWT, DWT, or WPT are known as wavelet analysis. The CWT is performed using a base function known as the basic wavelet function. CWT of the signal \(x(t)\) is defined as [31]:  
\[\text{CWT}_{x}(\tau,s)=\sqrt{s}\int_{-\infty}^{+\infty}X(f)\Psi^{*}(sf)e^{j2\pi f \tau}df \tag{1}\]  
where \(\psi(f)\) is the Fourier transform of \(\psi_{\tau,s}(t),\Psi^{*}(f)\) is the complex conjugate function of \(\psi(f)\). Wavelet transform alters signals function \(x(t)\) to become a bivariate function \(CWT_{x}(\tau,s)\). Parameter \(s\) performs to stretch \(\psi(t)\) function, and the parameter \(\tau\) performs to shift \(\psi(t)\) function on the time axis. The wavelet transform has mapped the \(x(t)\) signal function from a one-variate function to a bivariate function \(s\) and \(\tau\). By changing the scale coefficient \(s\) and shifting the base wavelet function along the time axis of a quantity \(\tau\), the wavelet coefficients \(|CWT_{x}(\tau,s)|\) may generate a three-dimensional graph representing the change of amplitude simultaneously along the time (matching \(\tau\)) and frequency axis (ratio with factor \(s\)). Several base wavelet functions \(\psi(t)\) are widely used in mechanical vibration analysis, such as Morlet-wavelet, Haar-wavelet, and Gabor-wavelet. In 2010, [PERSON] [32] introduced an improved algorithm called the wavelet-based synchrosqueezing transform.  
WSST generates a higher-resolution TFR of the signal compared to CWT. The theoretical basis of the WSST transformation is presented in detail in [33]. Subsequently, the generalized wavelet-based synchrosqueezing transform (GST) is a method that combines the generalized Fourier transform and the WSST transform. The main benefit of GST is its ability to identify the frequency of a signal changes over time within a narrow band, making it a promising tool for analyzing gear meshing vibrations in gearboxes with very close frequency bands [34; 35].  
This section reviewed the development of wavelet analysis from traditional methods like DWT and CWT to advanced methods such as WPT and WSST. These advancements have significantly improved signal resolution, making wavelet analysis a powerful tool for diagnosing faults in complex, non-stationary conditions. The continuous evolution of these techniques highlights their increasing importance in fault diagnostics.