## 3 The Proposed Features Extraction Method
### The Adopted Nonlinear Wavelet Synchrosqueezing and Its Reconstruction  
After the completion of the signal conversion frequency, the next step was to use the NWSST which mathematically can be described as a post-processing method of the wavelet transform (WT) to extract the fault information contained in the time-frequency plot. As a singular reallocation method, synchrosqueezing is based on continuous wavelet transform (CWT) and aims to refine the WT coefficient \(W(t,\omega)\) by transferring its value to different points \((t^{\prime},\omega^{\prime})\), from which IF lines and curves can be extracted. For a chosen mother wavelet function, the CWT of the HT signal \(z_{n}(t)\) is defined by:  
\[W_{x}(a,b)=\int\limits_{-\infty}^{+\infty}z_{n}(t)\frac{1}{\sqrt{a}}\psi( \frac{t-b}{a})dt \tag{49}\]  
where \(a\) is the scale factor, \(b\) the dilation factor, and \(\overline{\psi((t-b)/a)}\) represents the complex conjugate of \(\psi((t-b)/a)\). The mapping between scale factor \(a\) and signal frequency \(\omega\) makes it easier to visualise wavelet coefficients in the time-frequency plane. The Fourier transform \(\dot{\psi}(\xi)\) of the decayed mother wave function \(\psi\) is approximately equal to zero in the negative frequencies: \(\dot{\psi}(\xi)=0\) for \(\xi<0\) and is concentrated around \(a=\omega_{0}/\omega\). Its IF is suggested to be preliminarily evaluated by taking derivatives of wavelet coefficients. The formula for its computation is as follows:  
\[\omega_{x}(a,b)=\left\{\begin{array}{cl}-j\partial_{b}W_{x}(a,b)/W_{x}(a,b )&\text{if }\left|W_{x}(a,b)\right|>0\\ \infty&\text{if }\left|W_{x}(a,b)\right|<0\end{array}\right. \tag{50}\]  
With the map built by Equation (48), in the next synchrosqueezing step, the frequency variable \(\omega\) and scale factor \(a\) were computed only at discrete points \(a_{i}\), with \(\left(\Delta a\right)_{i}=a_{i}-a_{i-1}\), and its synchrosqueezing value was similarly determined only at the centres \(\omega_{s}\) of closed intervals \(\left[\omega_{s}-1/2\Delta\omega,\omega_{s}+1/2\Delta\omega\right]\), with \(\Delta\omega=\omega_{s}-\omega_{s-1}\). After obtaining the candidate IF, \(\omega_{s}(a,b)\), the reassignment operation of \(W_{x}(a,b)\), could be performed by summing these different contributions; the synchrosqueezing wavelet transform of \(x(t)\) was obtained as:  
\[T_{x}(\omega_{s},b)=\int\limits_{A(b)}W_{x}(a,b)a^{-3/2}\delta(\omega(a,b)- \omega)da \tag{51}\]  
The partial zoom of nonlinear synchrosqueezing of the time-frequency representation (TFR) can be mathematically described as constructing a time-frequency band-pass filter of the Hilbert transform \(H(\omega,b)\) for denoising. The partial zoom nonlinear synchrosqueezing TFR of the signal \(z_{n}^{*}(t)\) similar to Equation (49) was obtained from  
\[T_{z_{n}^{*}}(\omega_{s}^{*},b)=\int\limits_{A(b)}W_{x}^{*}(a,b)a^{-3/2}\delta( \omega^{*}(a,b)-\omega^{*})da \tag{52}\]  
where \(\omega^{*}(a,b)-\omega^{*}\) is the partial discrete frequency sequence of the partial zoom nonlinear synchrosqueezing TFR. The corresponding elements of matrices \(T_{zz}(\omega_{s}^{*},b)\) were then multiplied by \(H(\omega,b)\) to retain the components from the IF \(f_{i}\), and the fixed frequency bandwidth is 2D for time-frequency filtering as described in Equation (53).  
\[T_{z_{n}^{*}}^{*}(\omega_{s}^{*},b)=T_{z_{n}}(\omega_{s}^{*},b)\times H(\omega _{s}^{*},b) \tag{53}\]  
where the coefficients of the filter \(H(\omega,b)\) in this zone are 1 as proposed [15; 33], while the other time-frequency coefficients outside it are 0 for denoising, i.e., \(H(\omega,b)=\left\{\begin{array}{cc}1&(\omega,b)\in f_{i}\pm d\\ 0&(\omega,b)\
otin f_{i}\pm d\end{array}\right..\)  
The flow chart in Figure 4 describes the generic framework of the proposed method for multi-faults diagnostics of the coupled rotor system: