## Appendix A Conventional Instantaneous Frequencies Estimation Method
Above all, the estimated instantaneous frequencies can be expressed as  
\[v_{1}(t)=\text{Re}\bigg{[}\frac{1}{2\pi\Delta t}\frac{c(t)-c(t-\Delta t)}{i \cdot c(t)}\bigg{]}=\frac{1}{2\pi\Delta t}\frac{x(t-\Delta t)y(t)-x(t)y(t- \Delta t)}{x^{2}(t)+y^{2}(t)}\] (A15)  
which is considered the conventional instantaneous frequencies estimation method (CIF). Equation (A15) shows that the estimated instantaneous frequencies are affected by the time-sample interval, and precisions of estimated instantaneous frequencies are decreased with the increasing time-sample interval [21]; thus, CIF may lead to inaccurate instantaneous frequency estimation results. Appendix B High-Precision Instantaneous Frequencies Estimation Methods via Principal Instantaneous Phases  
Equation (A6) can be rewritten as  
\[v(t)=\frac{1}{2\pi}\frac{\partial\theta(t)}{\partial t}=\frac{1}{2\pi}\lim_{ \Delta t\to 0}\bigg{[}\frac{\theta(t)-\theta(t-\Delta t)}{\Delta t} \bigg{]}\] (A16)  
as parameter \(k\) in Equation (A3) cannot be determined, thus, \(\theta(t)\) cannot be directly used for instantaneous frequencies estimation; but if we can avoid period-jumps of \(\theta_{0}(t)\) and \(\theta_{1}(t)\), then they can be both directly used for instantaneous frequencies estimation. If \(\theta_{0}(t)\) is used for instantaneous frequencies estimation, then according to Equation (A16), the estimated instantaneous frequencies can be obtained by computing the derivative of arctangent function itself as [23]  
\[p_{1}(t)=\frac{1}{2\pi}\frac{\partial\theta_{0}(t)}{\partial t}=\frac{1}{2\pi }\frac{\partial\text{arctan}\big{[}x^{-1}(t)\cdot y(t)\big{]}}{\partial t}\] (A17)  
and  
\[p_{1}(t)=\frac{1}{2\pi}\frac{x(t)\frac{\partial y(t)}{\partial t}-y(t)\frac{ \partial x(t)}{\partial t}}{x^{2}(t)+y^{2}(t)}=v(t).\] (A18)  
Therefore, according to Equations (A9)-(A14), the estimated instantaneous frequencies can be expressed as  
\[p_{2}(t)=\frac{1}{2\pi\Delta t}\frac{x(t-\Delta t)y(t)-x(t)y(t-\Delta t)}{x^{ 2}(t)+y^{2}(t)}=v_{1}(t),\] (A19)  
and it means that this method can avoid phase period-jumps, but it cannot avoid the impact of the time-sample interval and may lead to inaccurate instantaneous frequencies estimation results. If \(\theta_{1}(t)\) is used for instantaneous frequencies estimation, then according to Equation (A16), the estimated instantaneous frequencies can be given by  
\[p_{3}(t)=\frac{1}{2\pi}\frac{\theta_{1}(t)-\theta_{1}(t-\Delta t)}{\Delta t}\] (A20)  
and it avoids the impact of the time-sample interval to obtain high-precision instantaneous frequencies, except for time-points corresponding to period-jumps of \(\theta_{1}(t)\). If we consider  
\[\Delta\theta_{1}(t)=\theta_{1}(t)-\theta_{1}(t-\Delta t)\] (A21)then  
\[-2\pi\leq\Delta\theta_{1}(t)\leq 2\pi.\] (A22)  
Based on Equation (A21), if we consider  
\[\Delta\theta(t)=\theta(t)-\theta(t-\Delta t)=2\pi\times\Delta t\times v(t),\] (A23)  
then according to the sampling theorem [26; 27], there should be  
\[-\pi\leq\Delta\theta(t)\leq\pi.\] (A24)  
Thus, we can modify \(\Delta\theta_{1}(t)\) into \(\Delta\theta_{2}(t)\) to meet the requirement expressed in Equation (A24) as  
\[\Delta\theta_{2}(t)=\left\{ \begin{array}{l}-\Delta\theta_{1}(t)-2\pi,\;\; \text{if}\qquad-2\pi\leq\Delta\theta_{1}(t)<\;-\pi\\ \Delta\theta_{1}(t),\;\;\text{if}\quad-\pi\leq\Delta\theta_{1}(t)\leq\pi\\ \Delta\theta_{1}(t)-2\pi,\qquad\text{if}\pi<\Delta\theta_{1}(t)\leq 2\pi \end{array} \right..\] (A25)  
Then  
\[-\pi\leq\Delta\theta_{2}(t)\leq\pi\] (A26)  
and estimated high-precision instantaneous frequencies can be given by  
\[h_{1}(t)=\frac{1}{2\pi}\frac{\Delta\theta_{2}(t)}{\Delta t}\] (A27)  
and it avoids impacts of time-sample interval and period-jumps of \(\theta_{1}(t)\).