# Wind Load Characteristics of Wind Barriers Induced by High-Speed Trains Based on Field Measurements
## 3 Data Processing Method
### Wavelet Analysis  
Wavelet transform (WT) has been widely applied as an excellent time-frequency tool to analyze non-stationary signals in many research aspects over the last decades. The WT can describe the signal time and frequency domain of the local characteristics. It is, therefore, an effective multi-resolution analysis method to identify wind pressure characteristics. WT can divide a signal into the low frequency component and the high-frequency component. This decomposition process can be iterated on the low-frequency component. Therefore, the original signal is decomposed into several levels of lower-resolution components. The reconstructed signal can possess the time characteristics of the original signal, and it can be used to filter the frequency distribution of the signal. By supposing \(\psi(t)\in L^{2}(R)\) (where \(L^{2}\) represents the Hilbert measurement space), the Fourier transform of \(\hat{\psi}(\omega)\), represented by \(\psi(t)\)', meets the wavelet admissible condition as follows:  
\[C_{\psi}=\int\limits_{R}\frac{\left|\hat{\psi}(\omega)\right|^{2}}{\left| \omega\right|}d\omega<\infty, \tag{2}\]  
where \(\psi(t)\) is the mother wavelet, \(R\) represents set of reals, and \(\omega\) represents the angular frequency. By extending and translating the mother wavelet, a wavelet family can be obtained as follows:  
\[\psi_{a,b}(t)=a^{-1/2}\psi\left(\frac{t-b}{a}\right), \tag{3}\]  
where \(a\) is the scale factor and \(b\) is the translation factor.  
For any function, \(f(t)\in L^{2}(R)\), the continuous wavelet transform of \(f(t)\), can be defined as follows:  
\[W_{f}(a,b)=a^{-1/2}\int\limits_{R}f(t)\sqrt[]{\psi}\left(\frac{t-b}{a}\right)dt. \tag{4}\]  
Actually, the continuous wavelet transform is highly redundant because of the continuous variation of the scale factor, \(a\), and the translation factor, \(b\). In practice, the scale factor, \(a\), and translation factor, \(b\), are usually discretized in order to improve computation efficiency. Therefore, the measurement is called a discretized wavelet transform and it can be described as follows:  
\[f(t)=\frac{1}{C_{\psi}}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{ \infty}\frac{1}{a^{2}}W_{f}(a,b)\psi\left(\frac{t-b}{a}\right)dadb. \tag{5}\]  
In the present study, the Db4 wavelet was selected for the discretized wavelet transform, which was adopted by [PERSON] et al. [30] to analyze aerodynamic loads on an overhead bridge due to the passage of high-speed trains. On the basis of binary sampling of the wavelet transform at all levels, the time history of the wind pressure can be decomposed to 13 stages. After the reconfiguration of scale factors and wavelet coefficients at different decomposition stages, the low-frequency part, \(a_{j}\) (approximate part), and the high-frequency part, \(d_{j}\) (detail part), of the original time-varying wind pressure at different scales could be obtained, as shown in Figure 5. The sampling frequency adopted in the process was \(F_{S}\) = 625 Hz. According to the Nyquist sampling theorem, the maximum frequency of the original wind pressure signal was \(F_{S}\)/2 = 312.5 Hz. The frequency band limits of the low- frequency parts and high-frequency parts at different scales were \(a_{j}\): (0, \(2^{-(j+1)}F_{S}\)) and \(d_{j}\): (\(2^{-(j+1)}F_{S}\), \(2^{-j}F_{S}\)) (\(j\) refers to the decomposition stage). The initial measured wind pressure at each point is defined to be zero before applying the wavelet transform. Therefore, the characteristics of train-induced wind pressure on the wind barrier's surface can be achieved through analyzing the results directly. This also indicates that the initial measured wind pressure at each point represents the natural wind before the train reaches the measurement site.