## 2 The Proposed Method and Theoretical Background
### Polynomial Frequency Modulation Integral Algorithm  
PCT is employed to better generate IF representation, and it can be defined as follows:  
\[\text{PCT}(t_{0},\omega,c(t))=\int_{-\infty}^{+\infty}s(t)\cdot w(t-t_{0}) \cdot e^{-j[\omega(t-t_{0})]}\cdot e^{-j[\omegaomega(t-t_{0})]}\cdot e^{-j[ \omegaomega(t-t_{0})]}\cdot e^{-j[\omegaomega(t-t_{0})]}\cdot e^{-j[\omegaomega (t-t_{0})]}\cdot e^{-j[\omegaomega(t-t_{0})]}\cdot e^{-j[\omegaomega(t-t_{0} )]}\cdot e^{-j[\omegaomega(t-t_{0})]}\cdot e^{-j[\omegaomega(t-t_{0})]} \tag{6}\]  
in which \(t_{0}\) represents the initial moment, \(s(t)\) represents the analytic signal with the TV parameter, \(w(t-t_{0})\) represents the window function, \(j=\sqrt{-1}\), \(\omega\) represents the IF, \(c\) represents the adaptive frequency demodulated coefficient, and \(e^{-j[\omegaomega(t-t_{0})]}\cdot e^{-j[\omegaomega(t-t_{0})]}\cdot e^{-j[ \omegaomega(t-t_{0})]}\) represents the TV demodulated operator. For the analytic signal \(s(t)\), it can be defined by amplitude \(a(t)\) and frequency \(\varphi(t)\):  
\[s(t)=a(t)e^{j[f(t)]\text{dt}} \tag{7}\]  
In a short period of time, \(\varphi(t)\) shows a linear variation regularity, and the corresponding Taylor expansion is given as follows:  
\[\varphi(t)=\varphi(t_{0})+\varphi^{(1)}(t_{0})(t-t_{0})^{1}+\cdots+\frac{ \varphi^{(i)}(t_{0})(t-t_{0})^{i}}{i!} \tag{8}\]  
in which \(\varphi^{(\cdot)}(t)\) represents the derivative of \(\varphi(t)\). According to the Equations (6)-(8), the PCT amplitude of IF \(\varphi(t)(\omega=\varphi(t))\) can be obtained as follows:  
Figure 2: The decomposition frame of multi-level KRRD. \[ \begin{array}{ll}|\text{PCT}(t_{0},\omega,c(t))|&=\left|\int_{-\infty}^{+ \infty}a(t)w(t-t_{0})e^{j\int\varphi(t)\text{dt}}e^{-j\left[\omega(t-t_{0}) \right]}e^{-j\sum_{i=1}^{n}\frac{c(t)(t-t_{0})^{j+1}}{j+1}}\text{dt}\right|\\ &=\left|\int_{-\infty}^{+\infty}a(t)w(t-t_{0})e^{j\left(\varphi(t)t_{0}+\cdots+ \varphi^{(i)}(t)(t-t_{0})^{i+1}/(i+1)\cdot\vec{t}\right)}e^{-j\varphi(t)t_{0} }e^{-j\sum_{i=1}^{n}\frac{c(t)(t-t_{0})^{j+1}}{j+1}}\text{dt}\right|\\ &=\left|\int_{-\infty}^{+\infty}e^{j\left(\sum_{i=1}^{n}\left(\frac{\varphi^{(i )}(t)}{\vec{e}}-c(t)\right)\right)}a(t)w(t-t_{0})\text{dt}\right|\\ &\leq\left|\int_{-\infty}^{+\infty}a(t)w(t-t_{0})\text{dt}\right|\end{array}  \tag{9}\]  
As shown in Equation (9), when \(\frac{\varphi^{(i)}(t)}{\vec{t}}\) is equal to \(c(t)\), the amplitude takes the maximum value. Meanwhile, the TF energy concentration is the highest, i.e., the optimal TF representation is obtained. In order to obtain the optimal demodulated operator, a novel parameter \(\alpha\) is introduced here [39]. The fitting angle in the TF domain is defined as \(\arctan(c)\), and the parameter \(\alpha\) can be defined by sampling time \(t_{\text{s}}\) and frequency \(f_{\text{s}}\) as follows:  
\[\alpha=\arctan\left|\frac{2t_{\text{s}}}{f_{\text{s}}}c\right|,\alpha\in\left( -\frac{\pi}{2},\frac{\pi}{2}\right) \tag{10}\]  
The discrete form of parameter \(\alpha\) and the update form of Equation (6) are shown as follows:  
\[\alpha=-\frac{\pi}{2}+\frac{\pi}{N_{\text{a}}+1},-\frac{\pi}{2}+\frac{2\pi}{N _{\text{a}}+1},\cdots,-\frac{\pi}{2}+\frac{N_{\text{a}}\pi}{N_{\text{a}}+1} \tag{11}\]  
\[\text{PCT}(t_{0},\omega,c(t))=\int_{-\infty}^{+\infty}s(t)\cdot w(t-t_{0}) \cdot e^{-j\left[\omega(t-t_{0})\right]}\cdot e^{-j\sum_{i=1}^{n}\frac{\max \left(f_{\text{s}}/2t_{\text{s}}\right)(t-t_{0})^{j+1}}{i+1}}\text{dt} \tag{12}\]  
The corresponding TF spectrum is given as follows:  
\[Spec(t_{0},\omega)=\left|\text{PCT}(t_{0},\omega,c_{m})\right|^{2} \tag{13}\]  
in which \(c_{m}\) is the discrete value corresponding to the maximum TF amplitude \([c_{i},\omega_{i}]\).