## 2 The Basic Theory
### Empirical Wavelet Transform  
In order to address the mode mixing or over-estimation phenomenon of the EMD method, [PERSON] [17] proposed an adaptive method named the empirical wavelet transform (EWT) method. The key step of EWT is segmenting the Fourier spectrum of the collected signal. EWT divides the spectrum into several portions, and each portion corresponds to a mode centered at the specific frequency and compact support, such as AM-FM signal. Assuming that we want to segment the Fourier spectrum into \(N\) continuous segments, it needs to find \((N-1)\) the largest local maxima in the Fourier spectrum. In most cases, \(0\) and \(\pi\) are always taken as two boundaries (the Fourier spectrum is normalized). For this set of maxima, we define that \(\omega_{n}\) is the boundary between each segment, where \(\omega_{0}=0\) and \(\omega_{N}=\pi\). Each segment is shown as \(A_{n}=[\omega_{n-1},\omega_{n}]\), then \(\cup_{n=1}^{N}A_{n}=[0,\pi]\). Centered on each \(\omega_{n}\), we define a transition phase \(T_{n}\), with width \(2\tau_{n}\) and \(\tau_{n}=\gamma\omega_{n};\gamma\) is properly chosen to get a tight frame and is given in Equation (9). \[\gamma<\min_{n}\!\left(\frac{\omega_{n+1}-\omega_{n}}{\omega_{n+1}+\omega_{n}}\right) \tag{9}\]  
According to the boundaries defined by the segmentation of the Fourier spectrum, a series of empirical wavelets can be constructed as bandpass filters based on the idea of the construction of both Littlewood-Paley and Meyer's wavelets [52]. For \(\forall n>0\), the empirical scaling function and the empirical wavelet function are defined by the following Equations (10) and (11), respectively. \[\hat{\phi}_{n}(\omega)=\left\{ \begin{array}{ll}1,&|\omega|\leq\omega_{n}- \tau_{n}\\ \cos\!\left[\frac{\pi}{2\beta}\!\left[\frac{1}{2\tau_{n}}[|\omega|-\omega_{n}+ \tau_{n}]\right]\right],&\omega_{n}-\tau_{n}\leq|\omega|\leq\omega_{n}+\tau_{n }\\ 0,&\mbox{otherwise}\end{array} \right. \tag{10}\]  
\[\hat{\psi}_{n}(\omega)=\left\{ \begin{array}{ll}1,&\omega_{n}+\tau_{n}\leq| \omega|\leq\omega_{n+1}-\tau_{n+1}\\ \cos\!\left[\frac{\pi}{2\gamma_{n+1}}(|\omega|-\omega_{n+1}+\tau_{n+1})\right],&\omega_{n+1}-\tau_{n+1}\leq|\omega|\leq\omega_{n+1}+\tau_{n+1}\\ \sin\!\left[\frac{\pi}{2\beta}\!\left(\frac{1}{2\tau_{n}}(|\omega|-\omega_{n}+ \tau_{n})\right)\right],&\omega_{n}-\tau_{n}\leq|\omega|\leq\omega_{n}+\tau_{n }\\ 0,&\mbox{otherwise}\end{array} \right. \tag{11}\]  
The function \(\beta(x)=x^{4}(35-84x+70x^{2}-20x^{3})\), which is the most commonly used [16]. Therefore, for \(\forall n>0\), Equations (10) and (11) can be simplified as follows:  
\[\hat{\phi}_{n}(\omega)=\left\{ \begin{array}{ll}1,&|\omega|\leq[1-\gamma \omega_{n}\\ \cos\!\left[\frac{\pi}{2\beta}\!\left[\frac{1}{2\gamma\omega_{n}}[|\omega|-[1 -\gamma]\omega_{n}]\right]\right],&\mbox{$[1-\gamma\omega_{n}\leq|\omega|\leq [1+\gamma\omega_{n}\\ 0,&\mbox{otherwise}\end{array} \right. \tag{12}\]  
\[\hat{\psi}_{n}(\omega)=\left\{ \begin{array}{ll}1,&(1+\gamma)\omega_{n}\leq| \omega|\leq(1-\gamma)\omega_{n+1}\\ \cos\!\left[\frac{\pi}{2}\beta\!\left(\frac{1}{2\gamma\omega_{n+1}}(|\omega|- (1-\gamma)\omega_{n+1})\right)\right],&(1-\gamma)\omega_{n+1}\leq|\omega|\leq (1+\gamma)\omega_{n+1}\\ \sin\!\left[\frac{\pi}{2}\beta\!\left(\frac{1}{2\tau_{n}}(|\omega|-\omega_{n}+ \tau_{n})\right)\right],&(1-\gamma)\omega_{n}\leq|\omega|\leq(1+\gamma)\omega _{n}\\ 0,&\mbox{otherwise}\end{array} \right. \tag{13}\]  
The EWT is defined in a similar way as the classic wavelet transform.