## 2 Principles and Methods
### High-Precision and High-Resolution SST via Time-Frequency Instantaneous Phases (HSST)  
According to Appendix B, we can use instantaneous phases of \(G(t,f)\) for high-precision time-frequency instantaneous frequency estimation and finally obtain high-precision and high-resolution SST results with high efficiency. If \(\theta(t,f)\) and \(\theta(t,f)\in(-\infty,+\infty)\) denote the argument angles of \(G(t,f)\), \(\theta_{1}(t,f)\) and \(\theta_{1}(t,f)\in[0,2\pi]\) denote the principal argument angles of \(G(t,f)\), then we have  
\[\left\{ \begin{array}{l}\theta(t,f)=2k\pi+\theta_{1}(t,f)\\ \theta(t_{2},f)>\theta(t_{1},f),\;\;if\;\;\;\;t_{2}>t_{1}\end{array} \right. \tag{7}\]  
where \(k\) denote integer numbers; \(\theta(t,f)\) can be considered time-frequency instantaneous phases of \(G(t,f)\); \(\theta_{1}(t,f)\) can be considered time-frequency principal instantaneous phases of \(G(t,f)\) given by  
\[\theta_{1}(t,f)=\left\{ \begin{array}{l}0.5\pi,ifx(t,f)=0,y(t,f)\geq 0\\ 1.5\pi,ifx(t,f)=0,y(t,f)<0\\ \theta_{0}(t,f),ifx(t,f)>0,y(t,f)\geq 0\\ \pi+\theta_{0}(t,f)ifx(t,f)<0,y(t,f)\geq 0\\ \pi+\theta_{0}(t,f),ifx(t,f)<0,y(t,f)<0\\ 2\pi+\theta_{0}(t,f)ifx(t,f)>0,y(t,f)<0\end{array} \right. \tag{8}\]  
and  
\[\theta_{0}(t,f)=\arctan\left[x^{-1}(t,f)\cdot y(t,f)\right]\in(-0.5\pi,0.5\pi) \tag{9}\]  
According to Equations (A6) and (A16), the time-frequency instantaneous frequencies of \(G(t,f)\) can be given by [22]  
\[V_{1}(t,f)=\frac{1}{2\pi}\frac{\partial\theta(t,f)}{\partial t}=\frac{1}{2\pi }\lim_{\Delta t\to 0}\left[\frac{\theta(t,f)-\theta(t-\Delta t,f)}{\Delta t}\right] \tag{10}\]  
but parameter \(k\) in Equation (7) cannot be determined; thus, \(\theta(t,f)\) cannot be used for time-frequency instantaneous frequencies estimation. If \(\theta_{0}(t,f)\) are used for time-frequency instantaneous frequencies estimation, according to Equations (A16)-(A19), \(V_{1}(t,f)\) in Equation (10) can be obtained by computing the derivative of the arctangent function itself to avoid period-jumps of \(\theta_{0}(t,f)\) as  
\[V_{3}(t,f)=\frac{1}{2\pi}\frac{\partial\theta_{0}(t,f)}{\partial t}=\frac{1} {2\pi}\frac{\partial\arctan\left[x^{-1}(t,f)\cdot y(t,f)\right]}{\partial t} \tag{11}\]  
and  
\[V_{3}(t,f)=\frac{1}{2\pi}\frac{x(t,f)\frac{\partial y(t,f)}{\partial t}-y(t, f)\frac{\partial x(t,f)}{\partial t}}{x^{2}(t,f)+y^{2}(t,f)}=V_{1}(t,f). \tag{12}\]  
Then, the estimated time-frequency instantaneous frequencies can be given by  
\[V_{4}(t,f)=\frac{1}{2\pi\Delta t}\frac{x(t-\Delta t,f)y(t,f)-y(t-\Delta t,f)x (t,f)}{x^{2}(t,f)+y^{2}(t,f)}=V_{2}(t,f) \tag{13}\]  
Thus, according to [21] and Figure 1, it avoids period-jumps of \(\theta_{0}(t,f)\), but it cannot avoid the impact of the time-sample interval. If \(\theta_{1}(t,f)\) is used for instantaneous frequencies estimation, then according to Equation (A21), the estimated time-frequency instantaneous frequencies can be given by  
\[V_{5}(t,f)=\frac{1}{2\pi}\frac{\theta_{1}(t,f)-\theta_{1}(t-\Delta t,f)}{ \Delta t}=\frac{1}{2\pi}\frac{\Delta\theta_{1}(t,f)}{\Delta t} \tag{14}\]and the corresponding SST results can be expressed as  
\[S_{3}(t,v)=\int G(t,f)df,if\quad V_{5}(t,f)\in[v-\Delta v,v+\Delta v] \tag{15}\]  
which is considered SST via the time-frequency principal instantaneous phases (PSST); PSST can avoid the impact of the time-sample interval to obtain high-resolution SST results, except for time-points corresponding to period-jumps of \(\theta_{1}(t,f)\). According to Equations (A21)-(A27), if we considered  
\[\Delta\theta_{2}(t,f)=\left\{ \begin{array}{l}-\Delta\theta_{1}(t,f)-2\pi, \;\;if\qquad-2\pi\leq\Delta\theta_{1}(t,f)<\;-\pi\\ \Delta\theta_{1}(t,f),\;if\quad-\pi\leq\Delta\theta_{1}(t,f)\leq\pi\\ \Delta\theta_{1}(t,f)-2\pi,\qquad if\pi<\Delta\theta_{1}(t,f)\leq 2\pi\end{array} \right.