## 2 The Proposed Method and Theoretical Background
### Polynomial Frequency Modulation Integral Algorithm
\([c_{i},\omega_{i}]\) is given as follows:  
\[[c_{i},\omega_{i}]=\arg\max\left|\text{PCT}(t_{0},\omega,c(t))\right| \tag{14}\]  
Based on Equation (14), the basis function \(h^{i}_{t_{i},\omega_{i},c_{i}}(t)\) of the \(i\)-th component signal can be described as follows:  
\[h^{i}_{t_{i},\omega_{i},c_{i}}(t)=w(t-t_{0})\cdot e^{j\left[\omega_{i}(t-t_{0} )+\sum_{i=1}^{n}\frac{c_{i}(t-t_{0})^{j+1}}{j+1}\right]} \tag{15}\]  
Then, the first and second derivatives of the basis function \(h^{i}_{t_{i},\omega_{i},c_{i}}(t)\) with respect to time are calculated as follows:  
\[ \begin{array}{ll}\overset{i}{h}_{t_{i},\omega_{i},c_{i}}^{i}(t)=w(t-t_{0}) \cdot\left[j\Big{(}\omega_{i}+c_{i}(t-t_{0})^{i}\Big{)}\right]\cdot e^{j\left[ \omega_{i}(t-t_{0})+\sum_{i=1}^{n}\frac{c_{i}(t-t_{0})^{j+1}}{i+1}\right]} \end{array}  \tag{16}\]  
\[ \begin{array}{ll}\overset{i}{h}_{t_{i},\omega_{i},c_{i}}^{i}(t)&=w(t-t_{0}) \cdot\left[j\dot{c}_{i}(t-t_{0})-\left(\omega_{i}+c_{i}(t-t_{0})^{i}\right)^{ 2}\right]\cdot e^{j\left[\omega_{i}(t-t_{0})+\sum_{i=1}^{n}\frac{c_{i}(t-t_{0}) ^{j+1}}{j+1}\right]}\\ &=w(t-t_{0})\cdot\left[j\dot{c}_{i}(t-t_{0})^{i-1}-\left(\omega_{i}+c_{i}(t-t_ {0})^{i}\right)^{2}\right]\cdot e^{j\left[\omega_{i}(t-t_{0})+\sum_{i=1}^{n} \frac{c_{i}(t-t_{0})^{j+1}}{j+1}\right]}\end{array}  \tag{17}\]In essence, PCT is still WT and supports signal reconstruction. In addition, the first derivative and the second derivative basis functions satisfy the admissible condition and decay condition [35], which are given as follows:  
\[C_{bi}=\int_{-\infty}^{+\infty}\frac{\left|\hat{h}^{i}(\omega)\right|^{2}}{ \omega}\mathrm{d}\omega<\infty \tag{18}\]  
\[h^{i}(\pm\infty)=\dot{h}^{i}(\pm\infty)=\dot{h}^{-i}(\pm\infty)=0 \tag{19}\]  
where \(\dot{h}^{i}(\omega)\) is the Fourier transform of the basis function. For the acceleration signal \(\ddot{x}(t)\), the corresponding velocity \(\dot{x}(t)\) and displacement signal \(x(t)\) are calculated through the once and twice integrations, which are defined as follows:  
\[\left\{ \begin{array}{l}\dot{x}(t)=\sum\limits_{l=1}^{N}\int\ddot{x}_{i}(t) \mathrm{dt}=\sum\limits_{i=1}^{N}\dddot{x}_{i}(t)\\ x(t)=\sum\limits_{i=1}^{N}\int\int\dddot{x}_{i}(t)\mathrm{dt}=\sum\limits_{i=1 }^{N}\dddot{x}_{i}(t)\end{array} \right. \tag{20}\]  
in which \(\dddot{\tau}\) and \(\dddot{\tau}\) represent the once and twice integrations, respectively. PCT based on \(\dot{h}^{i}_{t,\omega_{i},\dot{\
u}_{i}}(t)\) is applied to signal \(\dddot{x}_{i}(t)\), and the transform is carried out by the partial integration method. The corresponding computational process is shown as follows:  
\[ \begin{array}{l}\mathrm{PCT}_{\dot{h}^{i}_{t}}\Big{(}\dddot{x}_{i}(t)\Big{)} =\int_{-\infty}^{+\infty}\big{(}\int\dddot{x}_{i}(t)\mathrm{dt}\big{)}\dot{h} ^{-i}_{t_{0},\omega_{i},\dot{\
u}_{i}}(t)\mathrm{dt}\\ =\big{(}\int\dddot{x}_{i}(t)\mathrm{dt}\big{)}\dot{h}^{i}_{t_{0},\omega_{i}, \dot{\
u}_{i}}(t)\bigg{|}\begin{array}{l}+\infty\\ -\infty\end{array}-\big{(}\int_{-\infty}^{+\infty}\dddot{x}_{i}(t)\Big{)}\dot{ h}^{i}_{t_{0},\omega_{i},\dot{\
u}_{i}}(t)\mathrm{dt}\end{array}  \tag{21}\]  
Combined with Equation (19), the first term in the above equation is updated as follows:  
\[\Big{(}\int\dddot{x}_{i}(t)\mathrm{dt}\Big{)}\dot{h}^{i}_{t_{0},\omega_{i}, \dot{\
u}_{i}}(t)\bigg{|}_{-\infty}^{+\infty}=\Big{(}\int\dddot{x}_{i}(t) \mathrm{dt}\Big{)}\dot{h}^{i}_{t_{0},\omega_{i},\dot{\
u}_{i}}(+\infty)- \Big{(}\int\dddot{x}_{i}(t)\mathrm{dt}\Big{)}\dot{h}^{i}_{t_{0},\omega_{i}, \dot{\
u}_{i}}(-\infty)=0 \tag{22}\]  
Then, Equation (21) is updated as follows:  
\[\mathrm{PCT}_{\dot{h}^{i}}\Big{(}\dddot{x}_{i}(t)\Big{)}=-\bigg{(}\int_{- \infty}^{+\infty}\dddot{x}_{i}(t)\bigg{)}\dot{h}^{i}_{t_{0},\omega_{i},\dot{ \
u}_{i}}(t)\mathrm{dt}=-\mathrm{PCT}_{\dot{h}^{i}}\big{(}\dddot{x}_{i}(t) \big{)} \tag{23}\]  
Similarly, PCT based on \(\dot{h}^{i}_{t_{i},\omega_{i},\dot{\
u}_{i}}(t)\) is applied to signal \(\dddot{\overline{\dot{x}}}_{i}(t)\) and the corresponding equation is given as follows:  
\[