# Wavelet Time-Frequency Analysis on Bridge Resonance in Train-Track-Bridge Interactive System
## 1 Introduction
### Wavelet Time-Frequency Analysis Theory and Relevant Computational Deduction  
Similar to the Fourier transform, the trigonometric function is used as the basis function to compare the time-domain signal. On the premise of assuming \(\psi(t)\in L^{2}(R)\), the basis function \(\psi(t)\) in the wavelet transform meets the condition \(\int_{R}|\Psi(\omega)|^{2}|\omega|^{-1}\mathrm{d}\omega<\infty\), where the Fourier transform of \(\psi(t)\) is \(\Psi(\omega)\). The wavelet base \(\psi(t)\) is expanded and translated, and its expansion factor (also known as the scale factor) is \(a\), the translation factor is \(b\), and the function after translation and expansion is \(\psi_{a,b}(t)\), which is expressed as Equation (1) [17]. \[\psi_{a,b}(t)=\frac{1}{\sqrt{a}}\psi(\frac{t-b}{a}) \tag{1}\]  
For the signal \(x(t)\in L^{2}(R)\), its continuous wavelet transform (CWT) is expressed as Equation (2) [17]. \[WT_{x}(a,b)=a^{-0.5}\int_{-\infty}^{\infty}x(t)\psi^{*}(\frac{t-b}{a})\mathrm{ d}t \tag{2}\]  
Simultaneously, the inverse continuous wavelet transform (ICWT) to reconstruct the signal can be expressed as Equation (3) [17]. \[x_{\text{ICWT}}(t)=\left[\int_{R}|\Psi(\omega)|^{2}|\omega|^{-1}\mathrm{d} \omega\right]^{-1}\int_{0}^{\infty}\int_{-\infty}^{\infty}WT_{x}(a,b)\psi_{a,b }(t)\frac{\mathrm{d}a}{a^{2}}\mathrm{d}b \tag{3}\]  
Essentially speaking, the wavelet decomposition is to constantly change the center (real-time translation) and scale of the wavelet window, multiply it with the signal for integral operation, compress the scale at high frequency and stretch the scale at low frequency, so as to obtain the signal component at any time under each frequency scale. Generally, the CWT spectrum can be defined as scalogram in Equation (4), a type of signal energy distribution with the translation factor \(b\) and the scale factor \(a\). \[|WT_{x}(a,b)|^{2}=\left|a^{-0.5}\int_{-\infty}^{\infty}x(t)\psi^{*}(\frac{t-b} {a})\mathrm{d}t\right|^{2} \tag{4}\]  
According to [PERSON]'s theorem [18], the weighted integral of the amplitude square of wavelet transform on the scale-translation plane (\(a-b\)) is equal to the total energy of the signal in the time domain, shown in Equation (5). Therefore, the amplitude square of wavelet transform can be regarded as a representation of the time-frequency distribution of signal energy. \[\int_{-\infty}^{\infty}|x(t)|^{2}dt=\left[\int_{R}|\Psi(\omega)|^{2}|\omega|^{ -1}\mathrm{d}\omega\right]^{-1}\int_{0}^{\infty}\int_{-\infty}^{\infty}|WT_{x} (a,b)|^{2}\frac{da}{a^{2}}db \tag{5}\]  
The key to wavelet transform is the selection of wavelet basis function. The smaller the time domain window width is, the stronger the time domain analysis ability of the wavelet basis is. Similarly, the smaller the frequency domain window width, the stronger the frequency domain analysis ability of the wavelet base is. The bump wavelet [18] selected in this paper has small variance in the frequency domain and high frequency resolution, which can avoid energy leakage in the process of frequency conversion and be expressed in Equation (6) [19]. \[\Psi(\omega)=e^{(1-\frac{1}{1-e^{2}(\omega-y)^{2}})}\chi_{(\mu-\frac{1}{2},\mu+ \frac{1}{2})} \tag{6}\]  
where \(\chi\) is the indicator function, \(\mu\) and \(\sigma\) are the wavelet base parameters, and the effective interval \(\mu\) is [3; 6]. The effective interval of \(\sigma\) is [0.1, 1.2]. As for smaller \(\sigma\), the wavelet has good frequency resolution, but poor time resolution. As for larger \(\sigma\), it produces wavelets with better time resolution and poorer frequency resolution. Generally, in the program, the default values for \(\mu\) and \(\sigma\) are 5 and 0.6, respectively. The time history and frequency domain laws of the wavelet are shown in Figure 1.  
Moreover, the wavelet computational properties are of great significance to the establishment of motion equations in the time-frequency domain and mainly includes the linearity, the time translation, the derivation and the approximation.