## 2 Non-Parameterized Time-Frequency Analysis
### Wavelet Transform
#### 2.2.3 Strength and Weakness  
* Time-frequency localization and flexible multi-scale analysis
* Challenges in selecting wavelet basis and parameters  
The application of wavelet transform in the field of structural health monitoring (SHM) presents multiple advantages and disadvantages, with its strengths being particularly prominent: First, wavelets exhibit excellent time-frequency localization characteristics. This property enables wavelet transform to perform high-resolution analysis simultaneously in both the time and frequency domains, providing a unique advantage for processing non-stationary signals. In SHM, structural responses are often influenced by various factors, resulting in non-stationary signals. Wavelet transform can therefore more accurately capture the changing features of these signals, providing strong support for structural condition assessment. Second, wavelets facilitate multi-scale analysis. By adjusting thescale and position of the wavelet function, signals can be decomposed at different scales, thereby extracting hierarchical features. These features can reflect damage conditions and performance changes in the structure at different scales, providing rich information for SHM and damage identification. Furthermore, the flexibility of wavelets is another significant advantage. The diversity of wavelet basis functions allows for the selection of appropriate wavelet bases for transformation based on specific signal types and requirements. In the field of SHM, different structural types and damage forms may require different wavelet bases to better represent signal characteristics, and the flexibility of wavelets satisfies this need. Notably, the discrete wavelet transform (DWT) possesses high computational efficiency. This characteristic gives wavelet transform significant potential for real-time SHM. Especially in signal compression and feature extraction, DWT can efficiently process large amounts of data, providing rapid and accurate analysis results for the online monitoring of structures and generation of damage warnings. However, application of wavelet transform to the field of SHM also faces some challenges. First, the selection of wavelet basis functions is crucial. Different wavelet basis functions have significant impacts on analysis results, necessitating professional knowledge and insight to choose the appropriate ones. Regarding signal characteristics, it is necessary to fully take into account features such as the singularity, frequency components, and non-stationarity of the signal. Meanwhile, task requirements also play a decisive role. Different application scenarios, such as denoising, feature extraction, or fault location, impose diverse demands on wavelet bases. Additionally, the inherent properties of wavelet bases, including compact support, symmetry, vanishing moments, and regularity, are important factors that cannot be overlooked during the selection process. Common wavelet bases each have their own characteristics and applicable scopes. The Daubechies wavelet (dbN) exhibits compact support and high-order vanishing moments, enabling it to keenly capture the local singularities of signals. It performs remarkably well in the vibration signal analysis of mechanical faults in components like bearings and gears, and it has a distinct advantage in detecting transient signals (such as impact pulses). The Symlet wavelet (symN) is approximately symmetric, making it suitable for processing non-stationary signals. It plays a significant role in the early fault detection of rotor cracks and gears, and it can effectively preserve the phase information of signals during the signal decomposition process. As a continuous wavelet, the Morlet wavelet is widely applied in time-frequency analysis and is especially suitable for early fault detection, particularly the extraction of weak fault signals. It also offers high resolution for non-stationary signals. The Haar wavelet is characterized by simplicity and high computational efficiency.