## 2 Non-Parameterized Time-Frequency Analysis
### Wigner-Ville Distribution
#### 2.3.3 Strength and Weakness  
* High time-frequency resolution and energy concentration
* Precision in instantaneous frequency analysis
* Mathematical completeness and cross-term interference  
The advantages of the Wigner-Ville distribution (WVD) in structural health signal processing are evident.  
First, it possesses high time-frequency resolution and energy concentration [32]. The WVD directly reflects the distribution of signal energy in the time-frequency plane through the Fourier transform of the instantaneous correlation function, without the constraints of a window function, thereby breaking through the limitations of time-frequency resolution imposed by the short-time Fourier transform (STFT) and wavelet transform. For single-component linear frequency-modulated signals, such as the impulse response generated by structural damage, the time-frequency energy concentration of the WVD reaches theoretical optimality, enabling clear capture of high-frequency transient components, such as nonlinear vibrations resulting from crack propagation or bolt loosening [33].  
Furthermore, it exhibits precision in instantaneous frequency analysis [1]. The quadratic nature of the WVD allows it to directly represent changes in the instantaneous frequency of a signal, such as the time-varying characteristics of structural modal frequencies or frequency modulation phenomena in nonlinear vibrations. Compared to linear time-frequency methods (e.g., STFT), the WVD offers higher accuracy in the analysis of frequency-hopping signals and rapid frequency changes.  
Additionally, it is characterized by mathematical completeness and information richness. The WVD satisfies the marginal conditions (the integrals of time and frequency energy equal the total signal energy) and possesses phase shift invariance. The Fourier transform of its autocorrelation function can be directly used for modal parameter identification and damage localization. Moreover, the real-valued nature of the WVD simplifies the physical interpretation of the energy density distribution, making it suitable for energy decay analysis in structural health monitoring. Finally, it demonstrates adaptability as a non-parametric method. The WVD does not require pre-assuming a signal model, making it suitable for extracting unknown damage features under complex operating conditions. For example, in composite material delamination damage monitoring, the WVD can adaptively analyze the time-frequency characteristics of nonlinear acoustic emission signals without relying on a preset damage model [26,33].  
Of course, the WVD has limitations and challenges. First, there is the interference of cross-terms [25]. The bilinear nature of the WVD leads to the generation of spurious cross-terms in the time-frequency distribution of multicomponent signals (such as multimode vibrations or environmental noise interference). These cross-terms can obscure real damage features, potentially leading to misjudgment, especially in the presence of multiple coexisting types of damage or highly noisy environments.  
Second, there is the issue of non-positivity of energy and ambiguity in physical meaning. Although the WVD is a real-valued distribution, its energy density may exhibit negative values, conflicting with the concept of energy in a physical sense and limiting its application in quantitative energy analysis, such as damage severity assessment.  
Finally, there are limitations in computational complexity and real-time performance. The high resolution of the WVD relies on instantaneous correlation calculations over the entire data length, leading to low processing efficiency for long time series signals (such as long-term structural monitoring data) and making it difficult to meet real-time monitoring requirements.