## 2 Theory
### Synchrosqueezed Continuous Wavelet Transform  
This section presents the theoretical derivation of the forward and inverse transforms of SWT proposed by [PERSON] (2011) [29]. First, for a given signal \(f(t)\) in the time domain, we apply the CWT. Let the mother wavelet function selected be \(\varphi\); then, the transformation equation for CWT is as follows:  
\[W_{f}(a,b)=\int f(t)a^{-\frac{1}{2}}\varphi^{*}\left(\frac{t-b}{a}\right)dt \tag{3}\]  
In Equation (3), \(W_{f}(a,b)\) denotes the wavelet coefficients, where \(a\) is the scale parameter, \(b\) is the time parameter, and \(\varphi^{*}\) is the complex conjugate of the mother wavelet function. Commonly used mother wavelets include Haar, Morlet, Symlets, and Meyer wavelets. In this study, the Morlet wavelet is chosen due to its highly localized nature in the frequency domain ([PERSON] et al., 2013) [35]. According to the Plancherel theorem ([PERSON], 2002) [36], the beam-domain expression of Equation (3) is given by  
\[W_{f}(a,b)=\frac{1}{2\pi}\int\hat{f}(\varepsilon)a^{\frac{1}{2}}\hat{\phi}^{* }(a\varepsilon)e^{ib\varepsilon}d\varepsilon \tag{4}\]  
where \(\varepsilon\) is the angular frequency and \(\hat{f}(\varepsilon)\) and \(\hat{\phi}^{*}\) are the Fourier transforms of \(f(t)\) and \(\varepsilon(t)\), respectively. Furthermore, \(\hat{f}(\varepsilon)\) can be expressed as  
\[\hat{f}(\varepsilon)=\pi A[\delta(\varepsilon-\omega)+\delta(\varepsilon+ \omega)] \tag{5}\]  
By substituting Equation (5) into Equation (4), we obtain  
\[W_{f}(a,b)=\frac{A}{4\pi}a^{\frac{1}{2}}\hat{\phi}^{*}(a\varepsilon)e^{ib \varepsilon} \tag{6}\]  
where \(\hat{\phi}(\varepsilon)\) tends to zero in the negative frequency domain and is concentrated around \(\varepsilon=\omega_{0}\), and the coefficient \(W_{f}(a,b)\) becomes concentrated at \(a=\frac{\omega_{0}}{\omega}\) in the time-frequency domain and is distributed along the scale within a certain range. The estimated instantaneous frequency obtained by taking the partial derivative of the wavelet coefficients is given by  
\[\omega_{f}(a,b)=\left\{\begin{array}{cc}-i\Big{(}W_{f}(a,b)\Big{)}^{-1}\frac{ \partial}{\partial b}W_{f}(a,b),&\left|W_{f}(a,b)\right|>0\\ \infty,&\left|W_{f}(a,b)\right|>0\end{array}\right. \tag{7}\]  
Using Equation (7), the wavelet coefficients \(W_{f}(a,b)\) undergo a transformation from the time-scale domain to the time-frequency domain, resulting in \(W_{f}\Big{(}\omega_{f}(a,b),b\Big{)}\). In this domain, the compressed wavelet coefficient values \(T_{f}(\omega_{i},b)\) can be derived by compressing the values within the frequency interval \(\left[\omega_{i}-\frac{1}{2}\Delta\omega,\omega_{i}+\frac{1}{2}\Delta\omega\right]\) surrounding a given frequency \(\omega_{i}\). Consequently, the forward transformation formula for the synchronized wavelet transform is expressed as  
\[T_{f}(\omega_{i},b)=\Delta\omega^{-1}\sum_{a_{k}:|\omega_{k}(a_{k},b)-\omega_ {i}|\leq\frac{\Delta\omega}{2}}W_{f}(a_{k},b)a_{k}^{-\frac{3}{2}}\Delta a_{k} \tag{8}\]  
Similarly, the inverse transformation formula for SWT is given by  
\[f(t)=\text{Re}\Bigg{[}C_{\varphi}^{-1}\sum_{l}T_{f}(\omega_{i},b)(\triangle \omega)\Bigg{]} \tag{9}\]  
where \(C_{\varphi}^{-1}=\int_{0}^{\infty}\varphi^{*}(\varepsilon)\frac{d\varepsilon} {\varepsilon}\), Re denotes the real part, and \(\varphi^{*}(\varepsilon)\) is the Fourier transform of the conjugate of the mother wavelet function. The original signal can be approximately reconstructed from the inverse transform.