# Wavelet Time-Frequency Analysis on Bridge Resonance in Train-Track-Bridge Interactive System
## 1 Introduction
### Wavelet Time-Frequency Analysis Theory and Relevant Computational Deduction
Their definitions and deductions are listed as follows. As for linearity, \(x(t)\) and \(y(t)\) are supposed to be two signals linearly formulating signal \(z=kx+cy\) with constant coefficients \(k\) and \(c\). In this case, the wavelet transform of signal \(z\) can be expressed in Equation (7) by the linear superposition of the wavelet transforms of \(x\) and \(y\). \[WT_{z}(a,b)=a^{-0.5}\int_{-\infty}^{\infty}\left[kx(t)+cy(t) \right]\psi^{*}(\frac{t-b}{a})\mathrm{d}t \tag{7}\] \[=ka^{-0.5}\int_{-\infty}^{\infty}x(t)\psi^{*}(\frac{t-b}{a}) \mathrm{d}t+ca^{-0.5}\int_{-\infty}^{\infty}y(t)\psi^{*}(\frac{t-b}{a}) \mathrm{d}t\] \[=kWT_{x}(a,b)+cWT_{y}(a,b)\]  
As for the time translation, \(x(t)\) and \(y(t)\) are supposed to be two signals and \(y(t)\) can be defined by the \(x(t)\) with a delay factor \(\tau\), namely \(y(t)=x(t-\tau)\). In this case, the wavelet transform of signal \(y\) can be expressed in Equation (8) by the consistent translation for the translation factor \(b\) within the wavelet transform of \(x\). \[WT_{x}(a,b)=a^{-0.5}\int_{-\infty}^{\infty}x(t)\psi^{*}(\frac{t-b}{a})\mathrm{ d}t \tag{2}\]0 \[WT_{x}(a,b)=a^{-0.5}\int_{-\infty}^{\infty}x(t)\psi^{*}(\frac{t-b}{a})\mathrm{ d}t \tag{2}\]1 \[WT_{x}(a,b)=a^{-0.5}\int_{-\infty}^{\infty}x(t)\psi^{*}(\frac{t-b}{a})\mathrm{ d}t \tag{2}\]2  
\[WT_{x}(a,b)=a^{-0.5}\int_{-\infty}^{\infty}x(t)\psi^{*}(\frac{t-b}{a})\mathrm{ d}t \tag{2}\]3 Definition of bump wavelet: (**a**) History of bump wavelet; (**b**) Time–frequency spectra of bump wavelet. As for the derivation, \(x(t)\) and \(y(t)\) are supposed to be two signals and \(y(t)\) is the first-order derivative of time for \(x(t)\), namely \(y(t)=dx/dt\). In this case, the wavelet transform of signal \(y\) can be expressed in Equation (9) by the derivation of translation factor \(b\) within the wavelet transform of \(x\). \[ \begin{split}& WT_{y}(a,b)=a^{-0.5}\int_{-\infty}^{\infty}\frac{dx(t)} {dt}\psi^{*}(\frac{t-b}{a})\mathrm{d}t\\ &=\lim_{\Delta t\to 0}a^{-0.5}\int_{-\infty}^{\infty}\frac{x(t+ \Delta t)-x(t)}{\Delta t}\psi^{*}(\frac{t-b}{a})\mathrm{d}t\\ &=\lim_{\Delta t\to 0}\frac{WT_{x}(a\dot{b}+\Delta t)-WT_{x}(a \dot{b})}{\Delta t}\\ &=\frac{\partial WT_{x}(a\dot{b})}{\partial b}\end{split}  \tag{9}\]  
As for the approximation, \(x(t)\) and \(y(t)\) are supposed to be two signals formulating signal \(z=xy\) through the element-wise product with \(x(t)=A(t)\), namely, slowly varying the modulating function and \(y(t)=\exp(\dot{y}(t))\), namely the phase time-varying function. In this case, signal \(z\) can be regarded as an asymptotic signal if oscillation due to the phase term \(\varphi(t)\) is more than that contributed by the amplitude term \(A(t)\)[20]. Correspondingly, the wavelet transform of signal \(z\) can be approximated in Equation (10) by the multiplication of the wavelet transform of \(x\) and the original signal \(y\). \[ \begin{split}& WT_{z}(a,b)=a^{-0.5}\int_{-\infty}^{\infty}A(t) \exp(\dot{y}(t))\psi^{*}(\frac{t-b}{a})\mathrm{d}t\\ &=\sqrt{a}A(b)\exp(\dot{y}\dot{\varphi}(b))\Psi^{*}(a\dot{ \varphi}(b))+\mathrm{O}\|\dot{A},\dot{\phi}(b)\|\\ &\approx x(b)WT_{y}(a,b)\end{split}  \tag{10}\]