## 3 Wavelet Analysis for Gerbox Diagnostics
Using genetic algorithms, [PERSON] et al. [37; 40; 64; 65] improved WPT's decomposition, selected the optimal Daubechies wavelet and calculated the number of hidden neurons in the ANN network to accurately detect slight wear, medium wear, and broken gear teeth. [PERSON] [66] and their research group have developed an automatic system for detecting tooth cracks and broken teeth in gearboxes, utilizing DWT in conjunction with SVM. [PERSON] [67] proposed a method that combines DWT, genetic algorithms, and neural networks to detect tooth decay and cracks in the teeth automatically. [PERSON] and [PERSON] used artificial neural networks and wavelet packet transforms to identify gear faults [68]. [PERSON] and colleagues [69] combined feedforward neural networks and SVM using CWT to detect the local faults of gears in gearboxes early. Another research direction is effectively applying wavelet analysis in gearbox diagnosis, enhancing the quality of time-frequency distributions through resolution and representation. [PERSON] et al. [70] and [PERSON] et al. [71] employed time synchronous averaging (TSA) and CWT to detect faults in gear by analyzing characteristic components in the time-frequency distribution. [PERSON] [72] has represented the wavelet coefficients in polar coordinates to visualize better the periodic vibration signals for detecting a local fault of gears caused by meshing impact and fault elements in rolling bearings during operation. [PERSON] and [73] also utilized this type of graph to enhance the ability to detect tooth damage in cases when the gearbox operates at variable rotational speeds. [PERSON] and [PERSON] [34; 74] were the first authors to exploit the benefits of GST to create high-resolution time-frequency distributions within a narrow frequency range for identifying gear meshing frequency (GMF) and characteristic modulation sidebands related to tooth wear. Several studies have recently proposed complex analysis methods for the third group of methods. These include [PERSON] [75], who diagnosed the fault in gear shafts, and [PERSON] [76], who diagnosed gearbox faults in wind turbines. Some other researchers [77; 78; 79; 80] proposed new analysis methods for the early detection of gear faults in multi-stage gearboxes based on wavelet analysis. Alongside research on applying wavelet analysis in technical diagnostics, researchers continue to propose new forms of wavelet basis functions that are more suitable. [PERSON] and [PERSON] [81] used the \(t\)-test to validate a relationship between the feature values derived from the DWT coefficients and the vibration signals. Subsequently, the estimated probability obtained from the \(t\)-test was employed to make choices on wavelet functions. This encouraged research focused on developing novel wavelet functions for engineering applications. [PERSON] and colleagues [82] have introduced an innovative approach for calculating the DWT, where a wavelet function was immediately generated based on the statistical characteristics of the analyzed signal. A new family of base wavelets was proposed in [83] for detecting small changes in the vibration signal of a gear. A set of basic wavelet functions combined with multiple wavelet functions (multi-wavelet) has been applied to analyze oscillating signals to detect gearbox faults [84; 85; 86; 87; 88; 89]. This section examined the application of wavelet analysis for diagnosing gearbox faults, highlighting how wavelet coefficients from CWT, DWT, or WPT are used alongside mathematical and statistical methods. These approaches have proven effective in detecting and diagnosing faults under non-stationary conditions, showcasing the potential of wavelet analysis in enhancing diagnostic accuracy and reliability in industrial applications. Despite their effectiveness, these methods often struggle with noise and closely spaced frequency components. This underscores the need for new methods, providing better resolution and fault detection capabilities.