## 2 Basic Theory and Method  
### Empirical Wavelet Transform  
The empirical wavelet transform (EWT) divides the Fourier spectrum of a signal and constructs a wavelet filter bank to extract the intrinsic modes, and can be divided into three main components.  
1. Fourier spectrum segmentation of a signal  
Assuming that the Fourier support \([0,\pi]\) is segmented into contiguous \(N\) segments, then a total number of \(N+1\) boundaries is needed. According to the local maximum values, the Fourier spectrum of a signal is divided into \(N\) segments. Denote \(\omega_{n}\) to be the limits between each segments (where \(\omega_{0}=0\) and \(\omega_{n}=\pi\)). Each segment is represented as \(\Lambda_{n}=[\omega_{n-1},\omega_{n}]\). A transition phase \(T_{n}\) of width \(2\tau_{n}\) is then defined, as shown in Figure 1.  
2. Calculation of empirical scale function and empirical wavelet function  
The empirical wavelets are defined as bandpass filters on each \(\Lambda_{n}\). The empirical scale function \(\overset{\wedge}{\phi}(\omega)\) and empirical wavelet function \(\overset{\wedge}{\psi}(\omega)\) can be calculated by Equations (1) and (2), respectively.  
\[\overset{\wedge}{\phi}(\omega)=\left\{\begin{array}{c}\text{1 if}\ |\omega|\leq(1- \gamma)\omega_{n}\\ \cos\left[\frac{\pi}{2}\Big{(}\frac{1}{2\gamma\omega_{n}}(|\omega|-(1-\gamma) \omega_{n})\Big{)}\right]\\ \text{if}\ (1-\gamma)\omega_{n}\leq|\omega|\leq(1+\gamma)\omega_{n}\\ \text{0 otherwise}\end{array}\right. \tag{1}\]  
\[\overset{\wedge}{\psi}(\omega)=\left\{\begin{array}{c}\text{1 if}\ (1+\gamma) \omega_{n}\leq|\omega|\leq(1-\gamma)\omega_{n+1}\\ \cos\left[\frac{\pi}{2}\beta\Big{(}\frac{1}{2\omega_{n+1}}(|\omega|-(1-\gamma) \omega_{n+1})\Big{)}\right]\\ \text{if}\ (1-\gamma)\omega_{n+1}\leq|\omega|\leq(1+\gamma)\omega_{n+1}\\ \sin\left[\frac{\pi}{2}\beta\Big{(}\frac{1}{2\gamma\omega_{n+1}}(|\omega|-(1- \gamma)\omega_{n})\Big{)}\right]\\ \text{if}\ (1-\gamma)\omega_{n+1}\leq|\omega|\leq(1+\gamma)\omega_{n+1}\\ \text{0 otherwise}\end{array}\right. \tag{2}\]  
The ratio \(\gamma\) in Equations (1) and (2) is restricted to a small value as \(0\leq\gamma\leq\min_{n}[(\omega_{n+1}-\omega_{n})/(\omega_{n+1}+\omega_{n})]\) to ensure the empirical scaling function and the empirical wavelets are a tight frame of \(L^{2}(R)\). \(\omega_{n}:\tau_{n}=\gamma\omega_{n}\), \(0<\gamma<1\). The function \(\beta(x)\) is an arbitrary \(C^{k}([0,1])\) function, the most used \(\beta(x)\) is defined as:  
\[\beta(x)=x^{4}\Big{(}35-84x+70x^{2}+20x^{3}\Big{)} \tag{3}\]  
3. Calculation of approximation coefficients and the detail coefficients  
Figure 1: Partitioning of the Fourier axis.  
The approximation coefficients \(w_{f}^{\varepsilon}(0,t)\) and the detail coefficients \(w_{f}^{\varepsilon}(n,t)\) can be calculated by Equations (3) and (4), respectively.  
\[w_{f}^{\varepsilon}(0,t)=\left\langle f,\phi_{1}\right\rangle=\int f(\tau) \overline{\phi_{1}(\tau-t)}d\tau=\begin{pmatrix}\overset{\wedge}{f}(\omega) \overset{\wedge}{\phi_{1}}(\omega)\\ \end{pmatrix} \tag{4}\]  
\[w_{f}^{\varepsilon}(n,t)=\left\langle f,\psi_{n}\right\rangle=\int f(\tau) \overline{\psi_{n}(\tau-t)}d\tau=\begin{pmatrix}\overset{\wedge}{f}(\omega) \overset{\wedge}{\psi_{n}}(\omega)\\ \end{pmatrix}^{\vee} \tag{5}\]  
where \(\overset{\wedge}{\phi_{1}}(\omega)\) and \(\overset{\wedge}{\psi}(\omega)\) are defined by Equations (1) and (2), respectively.  
The inverse empirical wavelet transformation is carried out by the following equation:  
\[\begin{split}& f(t)=w_{f}^{\varepsilon}(0,t)\ast\phi_{1}(t)+ \overset{\wedge}{\underset{n=1}{\sum}}w_{f}^{\varepsilon}(n,t)\ast\psi_{n}(t) \\ &=\begin{pmatrix}\overset{\wedge}{w}_{f}^{\varepsilon}(0,\omega) \ast\overset{\wedge}{\phi_{1}}(\omega)+\overset{\wedge}{\underset{n=1}{\sum} }\overset{\wedge}{w}_{f}^{\varepsilon}(n,\omega)\ast\overset{\wedge}{\psi_{n} }(\omega)\\ \end{pmatrix}^{\vee}\end{split} \tag{6}\]