## 3 Feature Extraction Using Wavelet Packet Transform (WPT)  
Non-stationary signals are frequently encountered in a variety of engineering fields (e.g., wind, ocean, and earthquake engineering) [25]. The inability of conventional Fourier analysis to preserve the time dependence and describe the evolutionary spectral characteristics of non-stationary processes requires tools which allow time and frequency localization beyond customary Fourier analysis. The wavelet transform is used to decompose random processes into localized orthogonal basis functions, providing a convenient format for the modeling, analysis, and simulation of non-stationary processes [26]. The time and frequency analysis made possible by the wavelet transform provides insight into the character of transient signals through time-frequency maps of the time variant spectral decomposition that traditional approaches miss [25]. Wavelet packet component energy is an effective method to define and characterize a specific signal phenomenon in the time-frequency domain. [PERSON] and [PERSON]'s [27] demonstrated that the energy stored in a specific frequency band, at a certain level of wavelet packet decomposition, provides more potential for signal feature than the coefficients alone. [PERSON] and [PERSON] [28] utilized sensitivity analysis, to compare the four damage indices of frequency, modal, flexibility, and energy changes of wavelet packets. They subsequently concluded that the index based on wavelet packet energy had the best performance in detecting changes in elemental stiffness. Therefore, in this study, this method will be used for the feature extraction phase of the proposed method. The WPT of a time domain signal \(f(t)\) can be calculated using a recursive filter-decimation operation [29]. After \(j\)-levels of decomposition, the original signal \(f(t)\) can be expressed as:  
\[f(t)=\sum_{i=1}^{2j}f_{j}^{i}(t), \tag{1}\]  
\[f_{j}^{i}(t)=\sum_{i=1}^{2^{j}}C_{j,k}^{i}(t)\psi_{j,k}^{i}(t), \tag{2}\]  
Herein, a linear combination of wavelet functions \(\psi_{j,k}^{i}(t)\) can express the component signal \(f_{j}^{i}(t)\). Integers \(i,j\) and \(k\) are the parameters of the modulation, scale and translation, respectively; \(C_{j,k}^{i}(t)\) and \(\psi_{j,k}^{i}(t)\) are defined as the wavelet packet coefficients and the wavelet packet function. The wavelet packet coefficients can be obtained from. \[C_{j,k}^{i}=\int_{-\infty}^{\infty}f(t)\psi_{j,k}^{i}(t)dt, \tag{3}\]  
Frequency domain information is very important in damage detection of structures. Therefore, a high level of WPT is required to detect small changes in the signal. After  
Figure 1: Summary of non-probabilistic damage detection approach. WPT is carried out, the corresponding energies of these decomposed component signals can be used to assess the condition of the structure. The component energies of signal are expressed as follows:  
\[E_{j}^{i}=\int_{-\infty}^{\infty}f_{j}^{i}(t)^{2}dt, \tag{4}\]  
It can be proven that, when the mother wavelet is semi-orthogonal or orthogonal [30], the signal energy \(E_{f}\) is the summation of the \(j\)-level component energies as follows:  
\[E_{f}=\int_{-\infty}^{\infty}f^{2}(t)dt=\sum_{i=1}^{2^{j}}E_{j}^{i}, \tag{5}\]  
In this research, the relative energy corresponding to the component signals of the structural acceleration response has been used as WWLS-SVM input, thus, the relative energy \(E_{i}\) in \(i\)-frequency band can be expressed as:  
\[E_{i}=\frac{E_{j}^{i}}{E_{f}}, \tag{6}\]  
Battle-Lemarie is a symmetric wavelet basis function. In the frequency domain, this wavelet feature is a band filter, while the scale function is a lowpass filter. Consequently, the above two functions' frequency bands are overlapped in certain degree, indicating a desirable orthogonal feature [31]. Orthogonal wavelets represent the signal with as little information as possible, and without redundancy. These features are very useful in denoising and in matrix multiplications [32].