## 1 Introduction
[24] focus on the critical aspect of choosing the appropriate mother wavelet for particular applications. An in-depth tutorial on choosing wavelets based on signal properties was given, resulting in optimal analysis and accurate results. [PERSON] [25] investigates using complex wavelets to diagnose multiple faults in variable frequency drives. This paper emphasizes the advanced capabilities of complex wavelets in detecting detailed vibration indicators, leading to more precise fault diagnosis. Recent advancements in machine learning and artificial intelligence have significantly impacted the field of gearbox fault detection. [PERSON] et al. [26] proposed a multisensor data-fusion approach, combining linear predictive coefficients and mel-frequency cepstral coefficients with convolutional neural networks and long short-term memory networks for effective fault diagnosis, emphasizing the importance of addressing transient conditions in gearboxes. Additionally, [PERSON] et al. [27] developed a methodology involving statistical time feature reduction for diagnosing gearbox wear, highlighting the challenges of detecting faults during transient operations. Furthermore, [PERSON] et al. [28] investigated multisensor data fusion for gearbox fault diagnosis using a 2-D convolutional neural network and motor current signature analysis. This study demonstrated significant improvements in detecting transient fault conditions and the effectiveness of automatic feature extraction from motor current signals. Wavelet analysis is a potentially powerful diagnostic technique due to its ability to flexibly represent signals in the time-frequency domain with high resolution. However, its verification and application are limited to experimental models or simulated vibration signals. This work conducts a numerical analysis and experiments to validate the effectiveness of the recently updated wavelet analysis as a gear diagnostic tool. Overall, the article has specific contributions and novelty:  
* A thorough overview of the analysis of Wavelet and its variations: Update the publication on Wavelet analysis, focusing on the development of improving time-frequency resolution. Particularly, the application of Wavelet analysis in fault diagnostics in industrial gearboxes is noteworthy. * Construct a multi-component signal's mathematical model to identify gear faults' symptoms in a gearbox operating in non-stationary conditions: Propose a solution to identify fault symptoms via the time-frequency representation by integrating multiple wavelet analysis methods. * Construct a proposal method combining the latest wavelet transforms and other methods. * The proposed method provides the generation of input data sets for the neural network to autonomously classify faults, matching with the developments of the 4.0 industrial era. The remainder of the paper is structured as follows. Section 2 provides an overview of the development of wavelet analysis from discrete wavelet transforms to wavelet-based synchrosqueezing transforms and generalized wavelet-based synchrosqueezing transforms. Section 3 provides a review of works on the use of wavelet analysis to diagnose gearbox faults. The methodology is presented in Section 4, which provides the mathematical foundation of the proposed method. Section 5 describes the experimental test conducted to validate the effectiveness of the proposed method. Finally, the conclusion is presented in Section 6.