## 3 Methodology
### Synchrosqueezed Wavelet Transform  
[PERSON] [26] proposed a compromise method to cope with the failure of reconstruction through the Rearrangement Algorithm, that is, a Synchrosqueezing Algorithm that only squeezes along the frequency's direction. It can be regarded as a special case of the Rearrangement Algorithm, which preserves the phase information while improving the time-frequency resolution, thus realizing signal reconstruction.  
The synchrosqueezed wavelet transform assumes that the signal to be analyzed is multi-component. A multi-component signal is composed of a series of AM-FM signals, and its mathematical expression is shown in Formula (5) [55]:  
\[f(t)=\sum_{K=1}^{K}f_{k}(t)=\sum_{K=1}^{K}A_{k}(t)e^{i\phi_{k}(t)} \tag{5}\]  
where \(A_{k}(t)\) and \(\phi_{k}(t)\) can be regarded as instantaneous amplitude and instantaneous phase. When \(A_{k}(t)>0\), \(\phi^{\prime}_{k}(t)>0\), and \(\phi^{\prime}_{k+1}(t)>\phi^{\prime}_{k}(t)\) are simultaneously established, the ideal time-frequency characterization of \(f(t)\) is shown in Formula (6):  
\[TI_{f}(t,\omega)=\sum_{K=1}^{K}A_{k}(t)\delta\bigl{(}\omega-\phi^{\prime}_{k} (t)\bigr{)} \tag{6}\]  
If the mathematical formula of the signal \(f(t)\) is regarded as a single harmonic component with amplitude A, there are:  
\[f(t)=A\cos(\omega t) \tag{7}\]  
The following results can be produced based on continuous wavelet transform and the Plancherel Theorem:  
\[W_{f}(a,b)=\frac{1}{2\pi}\int\hat{f}(\xi)a^{\frac{1}{2}}\overline{\hat{\psi}( a\xi)}e^{ib\xi}d\xi=\frac{A}{4\pi}\int[\delta(\xi-\omega)+\delta(\xi+\omega)]a^{ \frac{1}{2}}\overline{\hat{\psi}(a\xi)}e^{ib\xi}d\xi \tag{8}\]  
where \(\hat{f}(\xi)\) and \(\hat{\psi}(\xi)\) are Fourier Transforms of the signal \(f(t)\) and wavelet function \(\psi\), respectively. If \(\hat{\psi}(\xi)\) satisfies \(\xi<0\), \(\hat{\psi}(\xi)=0\), the wavelet transform result \(f(t)\) should be concentrated on \(W_{f}(a,b)\). However, the actual wavelet transform result \(W_{f}(a,b)\) expands from \(a=\omega_{0}/\omega\) and the phenomena of instantaneous information ambiguity are produced. Therefore, when \(W_{f}(a,b)\
eq 0\) the instantaneous frequency will be:  
\[W_{f}(a,b)=-i(W_{f}(a,b))^{-1}\frac{\partial}{\partial b}\Bigl{(}W_{f}(a,b) \Bigr{)} \tag{9}\]  
The result of the operator is \(f(t)\) for the mathematical formula of a single harmonic signal \(W_{f}(a,b)=\omega\), while it is \(W_{f}(a,b)=W(a,b)\) for a weak modulation signal. The operator is the mapping operator of the synchrosqueezed wavelet transform.  
When coordinate mapping is performed on the wavelet coefficients of the continuous wavelet transform, the wavelet coefficients are transformed from time-scale domain to time frequency domain, that is, \(<a,b>\rightarrow<W_{f}(a,b),b>.\) In applications, \(a_{k}-a_{k-1}=\left(\Delta a\right)_{k}\) and \(\omega_{l}-\omega_{l-1}=\Delta\omega\), synchrosqueezed wavelet transform in the discrete state can be expressed as follows [56]:  
\[T_{f}(\omega_{l},b)=\left(\Delta\omega\right)^{-1}\sum_{a_{k}:|\omega(a_{k},b )-\omega_{l}|\leq\Delta\omega/2}W_{f}(a_{k},b){a_{k}}^{-\frac{3}{2}}\left( \Delta a\right)_{k} \tag{10}\]