# An Improved Empirical Wavelet Transform and Its Applications in Rolling Bearing Fault Diagnosis
## 2 Exposition of Traditional EWT Method  
### EWT Method  
The EWT method can be described by the following three steps: (1) adaptively segment the spectrum; (2) construct a suitable empirical wavelet filter bank according to the boundaries and filter the signal; and (3) reconstruct empirical modes and use Hilbert transform to demodulate and analyze. Step 1: Divide the boundaries in the frequency domain. The frequency domain of the signal is normalized to \([0,\pi]\) and divided into N frequency bands with different bandwidths. Let \(\omega\) be the boundary between the bands; the left boundary is \(\omega_{0}=0\), the right boundary is \(\omega_{\rm N}=\pi\), and the remaining boundaries can be expressed as \(\omega_{1},\omega_{2},\ldots,\omega_{\rm N}\), then each frequency band can be recorded as \(\Lambda_{\rm n}=[\omega_{\rm n-1},\omega_{\rm n}],\rm n=1,2,\ldots,\rm N\). It is obvious that \(\cup_{\rm n=1}^{\rm N}\Lambda_{\rm n}=[0,\pi]\). With each boundary as the center, a transition phase with a width of \(\rm T_{n}=2\tau_{n}\) is defined for building a window base. In this paper, the spectral trend estimation method based on the key function is used to obtain the boundary of each frequency band; instead of the method used by [PERSON], the midpoint of the adjacent maxima is used as the boundary condition, as shown in Section 3.1. Step 2: Construct an empirical wavelet filter bank. The Meyer wavelet was selected by [PERSON] as a basis function to construct an empirical wavelet. A group of mutually orthogonal trigonometric functions is designed in the transition phase where the boundary of the frequency band is located, and a constant is designed in the frequency band. The empirical scaling function \(\varnothing_{\rm n}(\omega)\) and the empirical wavelets \(\Psi_{\rm n}(\omega)\) can be defined as  
\[\widehat{\varnothing}_{\rm n}(\omega)= \begin{pmatrix}1;|\omega|\leq(1-\gamma) \omega_{\rm n}\\ \cos\biggl{[}\frac{\pi}{2}\beta\biggl{(}\frac{1}{2\gamma\omega_{\rm n}}(| \omega|-(1-\gamma)\omega_{\rm n})\biggr{)}\biggr{]};(1-\gamma)\omega_{\rm n} \leq|\omega|\leq(1+\gamma)\omega_{\rm n}\\ 0;\ \text{others}\end{pmatrix}  \tag{1}\]  
\[\widehat{\Psi}_{\rm n}(\omega)= \begin{pmatrix}1;(1+\gamma)\omega_{\rm n}\leq| \omega|\leq(1-\gamma)\omega_{\rm n+1}\\ \cos\biggl{[}\frac{\pi}{2}\beta\biggl{(}\frac{1}{2\gamma\omega_{\rm n+1}}(| \omega|-(1-\gamma)\omega_{\rm n+1})\biggr{)}\biggr{]};(1-\gamma)\omega_{\rm n +1}\leq|\omega|\leq(1+\gamma)\omega_{\rm n}\\ \sin\biggl{[}\frac{\pi}{2}\beta\biggl{(}\frac{1}{2\gamma\omega_{\rm n}}(| \omega|-(1-\gamma)\omega_{\rm n+1})\biggr{)}\biggr{]};\ (1-\gamma)\omega_{\rm n}\leq|\omega|\leq(1+\gamma)\omega_{\rm n}\\ 0;\ \text{others}\end{pmatrix}  \tag{2}\]  
where the transition function \(\beta(x)\), the coefficient \(\gamma\), and the transition phase \(\tau_{\rm n}\) are  
\[\beta(x)=x^{4}\Bigl{(}35-84x+70x^{2}-20x^{3}\Bigr{)} \tag{3}\]  
\[\gamma<min\biggl{(}\frac{\omega_{\rm n+1}-\omega_{\rm n}}{\omega_{\rm n+1}+ \omega_{\rm n}}\biggr{)} \tag{4}\]  
\[\tau_{\rm n}=\gamma\omega_{\rm n}\,\ 0<\gamma<1 \tag{5}\]  
Step 3: Conduct an empirical wavelet transform.