## 2 Principles
### Synchrosqueezing Generalized S-transform (SSGST)  
In this part, we give the basic principle of SSGST and derive the mathematical expressions of the positive transformation and lossless inverse transformation based on the mathematical theory. First, the Equation (5) can be reformulated as,  
\[ \begin{array}{ll}GST_{x}(f,b)&=A|f|\exp(-i2\pi f_{0}fb)\\ &\times\int_{-\infty}^{\infty}x(t)\,\exp\Bigl{\{}-\alpha[f(t-b)-\beta]^{2} \Bigr{\}}\exp[-i2\pi f_{0}f(t-b)]dt\end{array}  \tag{11}\]  
Let \(\psi(t)=A\exp[-\alpha(t-\beta)]\exp(i2\pi f_{0}t)\), then Equation (11) is expressed as,  
\[GST_{x}(f,b)=|f|\exp(-i2\pi f_{0}fb)\int_{-\infty}^{\infty}x(t)\overline{\psi [f(t-b)]}dt, \tag{12}\]  
where, \(\overline{\psi(t)}\) represents the complex conjugate of \(\psi(t)\). According to the Parseval theorem and transformation properties of scale and translation in Fourier Transform, the Equation (13) can be derived as follows:  
\[GST_{x}(f,b)=\frac{1}{2\pi}\exp(-i2\pi f_{0}fb)\int_{-\infty}^{\infty}\hat{x }(\omega)\,\overline{\hat{\psi}(f^{-1}\omega)}e^{ib\omega}d\omega, \tag{13}\]  
where, \(\hat{x}(\omega)\) denotes the Fourier Transform result of the signal \(x(t)\). \(\overline{\hat{\psi}(\omega)}\) is the complex conjugate of the Fourier Transform result of \(\psi(t)\). Especially, if \(\omega<0\), \(\hat{\psi}(\omega)=0\). Then, we can calculate the instantaneous frequency of the signal \(x(t)\) by using Equation (14):  
\[f_{x}(f,b)=f_{0}f+[i2\pi GST_{x}(f,b)]^{-1}\frac{\partial GST_{x}(f,b)}{ \partial b} \tag{14}\]Now, we testify the feasibility and rationality of Equation (14) using an example of a harmonic wave \(x(t)=A_{1}\cos 2\pi f_{1}t\). The Fourier transform result \(\hat{x}(\omega)\) of the signal \(x(t)\) is given as,  
\[\hat{x}(\omega)=A_{1}\pi[\delta(\omega-2\pi f_{1})+\delta(\omega+2\pi f_{1})], \tag{15}\]  
where \(A_{1}\) and \(f_{1}\) denote, respectively, the amplitude and frequency of the harmonic wave \(x(t)\). The GST of the harmonic wave \(x(t)\) can be calculated by substituting Equation (15) into Equation (13) and is given as,  
\[GST_{x}(f,b)=\frac{A_{1}}{2}\exp[-i2\pi(f_{0}f-f_{1})b]\overline{\hat{\psi}(2 \pi f^{-1}f_{1})} \tag{16}\]  
According to Equation (14), the instantaneous frequency of the harmonic signal \(x(t)\) is,  
\[ \begin{array}{ll}f_{x}(f,b)&=f_{0}f+[i2\pi GST_{x}(f,b)]^{-1}\frac{\partial GST _{x}(f,b)}{\partial b}\\ &=f_{0}f+\frac{iA_{1}\pi(f_{1}-f_{0})\times\exp[-i2\pi(f_{0}f-f_{1})b]\times \overline{\hat{\psi}(2\pi f^{-1}f_{1})}}{i2\pi\times\frac{A_{1}}{2}\times\exp [-i2\pi(f_{0}f-f_{1})b]\times\overline{\hat{\psi}(2\pi f^{-1}f_{1})}}=f_{1} \end{array}  \tag{17}\]  
Therefore, the instantaneous frequency can be estimated by Equation (14). SSGST is defined as the Equation (18) according to the theories of synchoseuzezing. \[SSGST_{x}(f_{l},b)=L_{f}^{-1}\sum_{f_{k}:[f_{l}(f_{k},b)-f_{l}]\leq d\pi/2}GST _{x}(f_{k},b)\exp(i2\pi f_{0}f_{k}b)f_{k}^{-1}\Delta f_{k}, \tag{18}\]  
where, \(f_{l}\) is the frequency of the SSGST result. \(L_{f}\) is the half length of frequency range \([f_{l}-L_{f},f_{l}+L_{f}]\) centered on the frequency point \(f_{l}\). \(f_{k}\) denotes the discrete frequency points in frequency ranges of the GST, and \(\Delta f_{k}=f_{k}-f_{k-1}\). The Equation (18) represents that the time-frequency spectra values among the frequency range \([f_{l}-L_{f},f_{l}+L_{f}]\) are superimposed on the frequency point \(f_{l}\), so that the SSGST has higher accuracy of time-frequency decomposition ability. Next, the inverse transform of SSGST is derived as follows:  
We integrate over frequency \(f\) on both sides of Equation (13), and obtain the Equation (19) by replacing \(f^{-1}\omega\) with \(\xi\).