## 2 Methodology
### Theory of EWT  
EWT is a new method of adaptive wavelet decomposition proposed by [PERSON] in 2013 [PERSON] (2013). The main idea is to extract different modes of a signal by constructing a set of bandpass filters. Figure 2 shows the workflow of the EWT method. The first step of the algorithm is to convert the signal from the time domain to the frequency domain by using the Fourier transform method. Subsequently, we segment the Fourier spectrum by detecting the boundaries automatically to construct a bandpass filter bank. Finally, the AM-FM components of the original signal centred on a specific frequency are obtained by filtering. Suppose the Fourier axis of a signal is normalised to \([0,2\pi]\). To follow [PERSON]'s theorem, the discussion scope is limited to \([0,\pi]\). The Fourier support interval \([0,\pi]\) is partitioned into N consecutive segments, and each segment is denoted as \(\Lambda_{n}=[\omega_{n-1},\omega_{n}]\), where \(\omega_{n-1}\) and \(\omega_{n}\) denote the boundary of each segment, with \(n=1,2\cdots N\), \(\omega_{0}=0\), \(\omega_{N}=\pi\), and \(\sum\limits_{n=1}^{N}\Lambda_{n}=[0,\pi]\). N consecutive segments, excluding \(\omega_{0}\) and \(\omega_{N}\), still need N-1 boundaries. The remaining boundaries of the divided Fourier spectrum can be determined according to the following:  
* Local maxima other than 0 and \(\pi\) should be found in the spectrum and arranged in descending order;
* If the number of local maxima \(M\) is less than \(N\), retain all the maxima and reset the number of Fourier segments \(N\) to \(M\). Otherwise, retain the first \(N\) maxima;
* Then, the \(N-1\) boundaries can be defined as the centre of two consecutive maxima. As shown in Figure 3, the transition zone is defined as the grey hatched area of width \(2\tau_{n}\), where \(\tau_{n}=\gamma\omega_{n}\) and \(0<\gamma<1\). We can ensure that there is no overlap between two consecutive transition zones when \(\gamma<\min_{n}(\frac{\omega_{n+1}-\omega_{n}}{\omega_{n+1}+\omega_{n}})\). The empirical wavelets are equivalent to bandpass filters with band width \(\Lambda_{n}\). The empirical scaling function \(\hat{\phi}_{n}(\omega)\) and the empirical wavelet function \(\hat{\psi}_{n}(\omega)\) can be expressed by (1) and (2), respectively, based on the idea of constructing [PERSON] and [PERSON]'s wavelets. \[\hat{\phi}_{n}(\omega)=\left\{ \begin{array}{ll}1,\,if\,|\omega|\leq\omega_{n }-\tau_{n}\\ \cos[\frac{\pi}{2}\hat{\beta}(\frac{1}{2\tau_{n}}(|\omega|-\omega_{n}+\tau_{n} ))],\,if\,\omega_{n+1}-\tau_{n+1}\leq\,\,\omega_{n+1}+\tau_{n+1}\\ 0,\,otherwise\end{array} \right. \tag{1}\]  
Figure 3: Segmentation of the Fourier spectrum. Figure 2: EWT workflow. \[\dot{\psi}_{n}(\omega)=\left\{ \begin{array}{ll}1,&if\omega_{n}+\tau_{n}\leq| \omega|\leq\omega_{n+1}-\tau_{n+1}\\ \cos[\frac{\pi}{2}\beta(\frac{1}{2\tau_{n+1}}(|\omega|-\omega_{n+1}+\tau_{n+1} ))],&if\omega_{n+1}-\tau_{n+1}\leq|\omega|\leq\omega_{n+1}+\tau_{n+1}\\ \sin[\frac{\pi}{2}\beta(\frac{1}{2\tau_{n}}(|\omega|-\omega_{n}+\tau_{n}))],& if\omega_{n}-\tau_{n}\leq|\omega|\leq\omega_{n}+\tau_{n}\end{array} \right. \tag{2}\]  
The function \(\beta(x)\) is an arbitrary \(C^{k}([0,1])\) function that meets the basic requirements, as in (3). \[\beta(x)=\left\{ \begin{array}{ll}0,&if\ x\leq 0\\ \beta(x)+\beta(1-x)=1\ \forall x\in[0,1]\\ 1,&if\ x\geq 1\end{array} \right. \tag{3}\]  
It has been proven in practice that many polynomials can satisfy this condition, although the polynomial proposed by [PERSON] used by [PERSON] is widely used, as shown in (4). \[\beta(x)=x^{4}(35-84x+70x^{2}-20x^{3}) \tag{4}\]  
EWT is analogous to the conventional WT. As shown in (5) and (6), the detail coefficients and approximation coefficients can be given by the inner product with the empirical wavelet function and empirical scaling function, respectively, where \(\wedge\) indicates Fourier transform, \(\vee\) indicates inverse Fourier transform, and \(-\) represents complex conjugation.