# Guided Wave-Based Monitoring of Evolution of Fatigue Damage in Glass Fiber/Epoxy Composites
## 3 Signal Processing and Feature Extraction  
For characterizing the accumulated internal fatigue damage, the first step is to extract damage-sensitive features from the sensed guided wave signals to indicate the presence and evolution of damage. The damage feature is a characteristic parameter or a set of parameters that could be obtained by signal processing techniques in time, frequency or time-frequency domains. Compared to traditional Fourier transform (FT)-based signal processing techniques, wavelet analysis is a widely used signal processing and feature extraction technique in guided wave-based SHM with its powerful capabilities for multiresolution analysis and time-frequency analysis [18; 19; 37; 38].  
Within the framework of wavelet analysis, a signal is decomposed into an approximation and a detail. The approximation is then itself split into a second-level approximation and detail, and the process is repeated [39]. For the sake of improvement, WPT decomposes not only the wavelet approximations at each level, but also the wavelet details. Therefore, WPT offers a richer analysis on the original signal and a complete binary decomposition tree can be produced. In this study, the guided wave signals obtained after different fatigue cycles are first decomposed by WPT into component signals to extract characteristic features, and then further analyzed to indicate the existence and evolution of internal fatigue damage.  
A wavelet packet (WP) is a linear combination of wavelet functions. It can be presented as the function \(\psi_{j,k}^{i}(t)\):  
\[\psi_{j,k}^{i}(t)=2^{j/2}\psi^{i}(2^{j}t-k) \tag{1}\]  
where \(i,j\), and \(k\) are the modulation, the scale, and the translation parameter, respectively. The wavelet \(\psi^{i}\) is obtained from a recursive filtering process as  
\[\psi^{2i}(t)=\sqrt{2}\sum_{k=-\infty}^{\infty}h(k)\psi^{i}(2t-k) \tag{2}\]  
\[\psi^{2i+1}(t)=\sqrt{2}\sum_{k=-\infty}^{\infty}g(k)\psi^{i}(2t-k) \tag{3}\]  
where the discrete filters \(h(k)\) and \(g(k)\) are quadrature mirror filters associated with the scaling function and the mother wavelet function, respectively.  
Figure 5: Response wave signals after different fatigue cycles for specimen S_B: (**a**) six signals and (**b**) zoomed signals.  
For a guided wave signal \(S(t)\), according to Equation (1), at each level \(j\), the WPT produces \(2^{j}\) component signals \(S_{j}^{i}(t)\) that are related to \(2^{j}\) sets of coefficients \(c_{j,k}^{i}\):  
\[S(t)=\sum_{i=1}^{2^{j}}S_{j}^{i}(t) \tag{4}\]  
\[S_{j}^{i}(t)=\sum_{k=-\infty}^{\infty}c_{j,k}^{i}\Psi_{j,k}^{j}(t) \tag{5}\]  
Each WP coefficient \(c_{j,k}^{i}\) is the inner product of the signal with the wavelet as  
\[c_{j,k}^{i}=\int_{-\infty}^{\infty}S(t)\psi_{j,k}^{i}(t)dt \tag{6}\]  
After WPT decomposition, the WP component energy related to a single component signal is equal to  
\[E_{j}^{i}=\int_{-\infty}^{\infty}S_{j}^{i}(t)^{2}dt \tag{7}\]  
which stands for energy stored in the component signal \(S_{j}^{i}(t)\).  
Based on the WP component signals, an energy vector containing WP component energies in multiple frequency bands is defined by Equation (8) in this study as a multivariate signal feature:  
\[\mathbf{E}=\left[\begin{array}{cccc}E_{j}^{1}&E_{j}^{2}&\cdots&E_{j}^{2^{j}- 1}&E_{j}^{2^{j}}\end{array}\right]^{T} \tag{8}\]  
It is used to characterize the occurrence and evolution of fatigue damage in the composite specimens. For real applications, the energies in some WP components are very small, thus the energy vector defined in Equation (8) can be further simplified by excluding these negligible components.