## 2 The Theory of the Proposed Method
### Empirical Wavelet Decomposition  
EWT is a new signal-processing algorithm to detect the different vibration modes based on the EMD method and wavelet analysis theory. It can effectively extract the different modes from a mixed vibration signal, by adaptively establishing an appropriate filter bank based on the Fourier spectrum. EMD can adaptively decompose a signal into IMFs and obtain a time-frequency resolution to extract the fault features, but it is lacking the supporting mathematical theory. In contrast, EWT has no frequency resolution, but also a clear mathematical theoretical basis and the high computation speed of the discrete WT. The EWT procedure is described as follows:  
Firstly, accordingly to the Shannon criteria, the frequency limit is defined as \([0,\pi]\) in a normalized Fourier axis of periodicity \(2\pi\). The initial Fourier support \([0,\pi]\) is segmented into N contiguous segments containing the spectra of definite modes. The boundary of each segment is denoted as \(\omega_{n}\), and hence, the bandpass filter is obtained: \(\Lambda_{n}=[\omega_{n-1},\omega_{n}]\). In order to realize the EWT on the segmental support \(\Lambda_{n}\), the transition phase \(T_{n}\) of width \(2\tau_{n}\) is defined as being centered on each \(\omega_{n}\). An empirical filter bank is provided in Figure 2. The simplest \(\tau_{n}\) is proportional to \(\omega_{n}\tau_{n}=\gamma\omega_{n}\), where the range of \(\gamma\) is \(0<\gamma<\min_{n}\frac{\omega_{n+1}-\omega_{n}}{\omega_{n+1}+\omega_{n}}\). Next, the empirical wavelets are defined as bandpass filters on each Fourier support \(\Lambda_{n}\). The empirical scaling function and empirical wavelets are established using the idea of the construction of both Littlewood-Paley and [PERSON]'s wavelets, respectively. The simplified form is shown:  
\[\overset{\wedge}{\phi}_{n}(\omega)= \begin{cases}1&\big{|}\omega\big{|} \leq(1-\gamma)\omega_{n}\\ \cos[\frac{\pi}{2}\beta(\frac{1}{2\gamma\omega_{n}}(\big{|}\omega\big{|}-(1- \gamma)\omega_{n}))]&(1-\gamma)\omega_{n}\leq\big{|}\omega\big{|}\leq(1+\gamma )\omega_{n}\\ 0&\text{otherwise}\end{cases}  \tag{7}\]  
\[\overset{\wedge}{\psi}_{n}(\omega)= \begin{cases}1&(1+\gamma)\omega_{n}\leq \big{|}\omega\big{|}\leq(1-\gamma)\omega_{n+1}\\ \cos[\frac{\pi}{2}\beta(\frac{1}{2\gamma\omega_{n+1}}(\big{|}\omega\big{|}-(1 -\gamma)\omega_{n+1}))]&(1-\gamma)\omega_{n+1}\leq\big{|}\omega\big{|}\leq(1+ \gamma)\omega_{n+1}\\ \sin[\frac{\pi}{2}\beta(\frac{1}{2\gamma\omega_{n}}(\big{|}\omega\big{|}-(1- \gamma)\omega_{n}))]&(1-\gamma)\omega_{n}\leq\big{|}\omega\big{|}\leq(1+\gamma )\omega_{n}\\ 0&\text{otherwise}\end{cases}  \tag{8}\]  
where \(\beta(x)\) is an arbitrary function with an independent variable \(x\):  
\[\beta(x)= \begin{cases}0&x\leq 0\\ x^{4}(35-84x+70x^{2}-20x^{3})&0<x<1\\ 1&x\geq 1\end{cases}  \tag{9}\]  
According to the theoretical framework of the wavelet transform, the empirical wavelet transform is built by the constructed bandpass wavelet filter bank. The detail coefficient of the EWT is the inner product between the analyzed signal and the empirical wavelet in the time domain, and is the inverse Fourier transform of the of the frequency function production:  
\[W_{s}^{c}(n,t)=\big{\langle}s(t),\psi_{n}(t)\big{\rangle}=\int s(\tau) \overline{\psi_{n}(\tau-t)}d\tau=\mathcal{F}^{-1}\Bigg{(}\overset{\wedge}{ \widetilde{S}}(\omega)\overline{\overset{\wedge}{\psi}_{n}(\omega)}\Bigg{)} \tag{10}\]  
The approximation coefficient of the EWT is the inner product between the analyzed signal and the scaling function in the time domain, and is the inverse Fourier transform of the frequency function production:  
\[W_{s}^{c}(0,t)=\Big{\langle}s(t),\phi_{1}(t)\Big{\rangle}=\int s(\tau) \overline{\phi_{n}(\tau-t)}d\tau=\mathcal{F}^{-1}\Bigg{(}\overset{\wedge}{ \widetilde{S}}(\omega)\overline{\overset{\wedge}{\phi}_{n}(\omega)}\Bigg{)} \tag{11}\]  
Finally, the mono-component modes are obtained through the details and approximation coefficient of the EWT.