## 2 Principles
### Synchrosqueezed S Transform (SSST)  
The ST of \(x(t)\) is defined in Equation (1). Thus, the following is true:  
\[\psi(t)=\frac{1}{\sqrt{2\pi}}exp\bigg{(}\frac{t^{2}}{2}\bigg{)}exp(i2\pi t) \tag{4}\]  
According to the Parseval theorem, Equation (1) can also be expressed as follows:  
\[ST_{x}(f,b)=\frac{1}{2\pi}exp(-i2\pi fb)\int_{-\infty}^{+\infty}X(\xi)\overline {\psi(f^{-1}\xi)}exp(ib\xi)\mathrm{d}\xi \tag{5}\]  
where \(\xi\) is the angular frequency, \(X(\xi)\) denotes the FT result of the signal \(x(t)\), and \(\overline{\psi(\xi)}\) denotes the complex conjugate of the FT result of \(\psi(t)\). In particular, if \(\xi<0\), \(X(\xi)=0\).  
For a purely harmonic wave \(x(t)=A\cos(2\pi f_{0}t)\), its FT is presented as follows:  
\[X(\xi)=A\pi[\delta(\xi-2\pi f_{0})+\delta(\xi+2\pi f_{0})] \tag{6}\]  
where \(A\) and \(f_{0}\) denote, respectively, the amplitude and frequency.  
Then, we can obtain the instantaneous frequency of the signal \(x(t)\) as follows:  
\[\frac{\partial}{\partial\overline{b}}ST_{x}(f,b)=-i\pi A(f-f_{0})exp(-i2\pi(f-f_{ 0})b)\overline{\psi(2\pi f^{-1}f_{0})} \tag{7}\]  
\[\widetilde{f}(f,b)=f+[i2\pi ST_{x}(f,b)]^{-1}\frac{\partial}{\partial b}ST_{x} (f,b). \tag{8}\]  
For the harmonic wave \(x(t)\), the instantaneous frequency can be obtained by the following:  
\[\widetilde{f}(f,b)=f_{0} \tag{9}\]  
that is, the instantaneous frequency of the standard sinusoidal signal is the frequency of the signal itself.  
Given the candidate instantaneous frequency \(\widetilde{f}(f,b)\) at each location \((f,b)\) on the spectrogram, the TF spectrogram now becomes the time-instantaneous frequency spectrogram, i.e., locations \((f,b)\) are mapped to (\(\widetilde{f}(f,b)\),\(b\)) [21]. The next step is to reassign (squeeze) the value of the time-instantaneous frequency spectrogram.  
Note that the \(ST_{x}(f,b)\) is only computed at discrete values \(f_{k}\), and frequencies on the \"squeezed\" TF spectrogram are \(\widetilde{f}_{I}\). Hence, we squeeze the TF coefficients \(ST_{x}(f,b)\) together via the same instantaneous frequency \(\widetilde{f}(f,b)\). By superimposing the time-frequency spectra values among the frequency range \(\left[\widetilde{f}_{I}-\frac{\Delta\widetilde{f}_{I}}{2},\widetilde{f}_{I}+ \frac{\Delta\widetilde{f}_{I}}{2}\right]\) on the frequency point \(\widetilde{f}_{I}\), we then obtain a sharpened TF representation [22; 23], thus getting the SSST. For signal \(x(t)\), the SSST [24] is defined as follows:  
\[SSST_{x}(\widetilde{f}_{I},b)=\left(\Delta\widetilde{f}_{I}\right)^{-1}\times \sum_{f_{k}:|\widetilde{f}_{I}(f_{k},b)-\widetilde{f}_{I}|\leq\frac{1}{2} \Delta\widetilde{f}_{I}}|ST_{x}(f_{k},b)|f_{k}\Delta f_{k} \tag{10}\]  
where \(f_{k}\) is the discrete frequency obtained by the ST; \(\Delta f_{k}=f_{k}-f_{k-1}\); \(\widetilde{f}_{I}\) denotes the center frequency of the SSST; \(\Delta\widetilde{f}_{I}\) denotes the bandwidth; and \(\Delta\widetilde{f}_{I}=\widetilde{f}_{I}-\widetilde{f}_{I-1}\). Equation (10) shows that the SSST has higher accuracy in its TF decomposition ability.  
Similar to the ST, the signal can be reconstructed from the \(SSST_{x}\)[25]. The signal can be reconstructed as follows:  
\[x(b)=RE[(C_{\psi}C_{\varphi})^{-1}\sum_{I}SSST_{x}(\widetilde{f}_{I},b)\Delta \widetilde{f}_{I}] \tag{11}\]  
where \(RE\) means the real part and  
\[C_{\psi}=0.5\int_{0}^{+\infty}-\overline{\widetilde{\psi}(\xi)}\xi^{-1}\text {d}\xi,\ C_{\varphi}=\exp(-i[2\pi fb+\varphi(f,b)])f^{2} \tag{12}\]  
Details of the derivation can be found in the Appendix A.