## Appendix A Conventional Instantaneous Frequencies Estimation Method  
For a complex signal  
\[c(t)=x(t)+iy(t) \tag{1}\]  
where \(x(t)\) and \(y(t)\) are both real signals; if \(A(t)\) and \(A(t)\in[0,+\infty)\) denote argument amplitudes of \(c(t)\), \(\theta(t)\) and \(\theta(t)\in(-\infty,+\infty)\) denote argument angles of \(c(t)\),\(\theta_{1}(t)\) and \(\theta  
As principal periods (or period-jumps) of \(\theta_{0}(t)\) and \(\theta_{1}(t)\) are \(\pi\) and \(2\pi\), respectively; thus, \(\theta_{0}(t)\) and \(\theta_{1}(t)\) are both discontinuous functions, and their derivatives do not exist. Therefore, instantaneous frequencies of \(c(t)\) can be given by [22; 23]  
\[v(t)=\frac{1}{2\pi}\frac{\partial\theta(t)}{\partial t} \tag{10}\]  
If taking natural logarithm to Equation (11), we have  
\[i\theta(t)=\ln[c(t)]-\ln[A(t)]. \tag{11}\]  
Then, Equation (10) can be expressed as  
\[v(t)=\text{Re}\left[\frac{1}{2\pi i}\frac{1}{c(t)}\frac{\partial c(t)}{ \partial t}\right]=\frac{1}{2\pi}\frac{x(t)\frac{\partial y(t)}{\partial t}-y (t)\frac{\partial x(t)}{\partial t}}{x^{2}(t)+y^{2}(t)}. \tag{12}\]  
Usually, \(x(t)\) and \(y(t)\) are both implicit, and we cannot directly obtain their derivatives; but, if Fourier transform [22] results of \(c(t)\), \(x(t)\), and \(y(t)\) all exist, we can use them to obtain the corresponding derivatives; it means that if  
\[\left\{ \begin{array}{l}0\leq\int_{-\infty}^{+\infty}x^{2}(t)dt<+\infty\\ 0\leq\int_{-\infty}^{+\infty}y^{2}(t)dt<+\infty\end{array} \right. \tag{13}\]  
then  
\[\left\{ \begin{array}{l}\frac{\partial c(t)}{\partial t}=c_{0}(t)\\ \frac{\partial x(t)}{\partial t}=x_{0}(t)=\text{Re}[c_{1}(t)]\\ \frac{\partial y(t)}{\partial t}=y_{0}(t)=\text{Re}[c_{2}(t)]\end{array} \right. \tag{14}\]  
and  
\[\left\{ \begin{array}{l}c_{0}(t)=\int\left[i2\pi\eta\cdot C(\eta)\right] \cdot\exp[i2\pi\eta t]d\eta\\ c_{1}(t)=\int\left[i2\pi\eta\cdot X(\eta)\right]\cdot\exp[i2\pi\eta t]d\eta\\ c_{2}(t)=\int\left[i2\pi\eta\cdot Y(\eta)\right]\cdot\exp[i2\pi\eta t]d\eta\end{array} \right. \tag{15}\]  
where \(\text{Re}[\,]\) takes the real part of a complex number; \(c_{0}(t)\),\(c_{1}(t)\), and \(c_{2}(t)\) are derivatives of \(c(t)\),\(x(t)\), and \(y(t)\), respectively; \(\eta\) is the frequency; \(C(\eta)\), \(X(\eta)\), and \(Y(\eta)\) denote the Fourier transform results of \(c(t)\), \(x(t)\), and \(y(t)\), respectively; thus, Equation (12) can be rewritten as  
\[v(t)=\text{Re}\bigg{[}\frac{1}{2\pi i}\frac{c_{0}(t)}{c(t)}\bigg{]}=\frac{1}{ 2\pi}\frac{x(t)\cdot y_{0}(t)-y(t)\cdot x_{0}(t)}{x^{2}(t)+y^{2}(t)}. \tag{16}\]  
Due to impacts of dynamic range [9; 21] and amplitude amplified scale factor \(2\pi\eta\), background noises beyond effective bandwidths [9; 21] of \(C(\eta)\), \(X(\eta)\), and \(Y(\eta)\) will seriously affect the stabilities of \(c_{0}(t)\), \(c_{1}(t)\), and \(c_{2}(t)\), respectively, and finally reduce the stabilities and precisions of \(v(t)\). Usually, we can replace derivatives \(c_{0}(t)\), \(c_{1}(t)\), and \(c_{2}(t)\) in Equation (16) with their corresponding differences; thus, estimated instantaneous frequencies can be expressed as  
\[v_{1}(t)=\text{Re}\bigg{[}\frac{1}{2\pi i}\frac{d_{0}(t)}{c(t)}\bigg{]}=\frac {1}{2\pi}\frac{x(t)\cdot d_{2}(t)-y(t)\cdot d_{1}(t)}{x^{2}(t)+y^{2}(t)} \tag{17}\]  
and  
\[\left\{ \begin{array}{l}d_{0}(t)=\frac{c(t)-c(t-\Delta t)}{\Delta t}\\ d_{1}(t)=\frac{x(t)-x(t-\Delta t)}{\Delta t}\\ d_{2}(t)=\frac{y(t)-y(t-\Delta t)}{\Delta t}\end{array} \right. \tag{18}\]  
where \(\Delta t\) denotes the time-sample interval. If \(x(t)\) and \(y(t)\) do not meet the requirements expressed in Equation (A9), such as \(x(t)=t^{2}\) and so on, then \(x(t)\) and \(y(t)\) do not have Fourier transform results; therefore, we cannot obtain derivatives \(c_{0}(t)\), \(c_{1}(t)\), and \(c_{2}(t)\), but we can replace them in Equation (A12) with their corresponding differences. Correspondingly, the estimated instantaneous frequencies can be expressed as Equations (A13) and (A14).