# An Optimal Resonant Frequency Band Feature Extraction Method Based on Empirical Wavelet Transform
## 2 Theoretical Background  
### Empirical Wavelet Transform  
The EWT is discussed under the Shannon-Nyquist criterion. First, it is assumed that the Fourier spectrum range is normalized to \([0,\pi]\); then, the signal of the Fourier spectrum adaptive segmentation is set to represent different modes of \(N\) interval, and the required number of boundaries is \(N+1\), which is constructed as a set of filter banks, where each segment can be recorded as \(\Lambda_{n}=[\omega_{n-1},\omega_{n}]\), where \(\omega_{0}=0\) and \(\omega_{n}=\pi\), i.e., \(\sum_{n=1}^{N}\Lambda_{n}=[0,\pi]\). Thus, an empirical wavelet function with orthogonal properties is obtained. Finally, the AM-FM single component with an orthogonality property of the Fourier spectrum can be extracted by a wavelet transformation of the segmented interval in the frequency domain.  
After determining the segmentation interval \(\Lambda\)n, the empirical wavelet is defined as the bandpass filter on each \(\Lambda\)n. [PERSON] et al. adopts the construction method of [PERSON]'s wavelet. The empirical scale function \(\hat{\phi}_{n}\) and the empirical wavelet \(\hat{\psi}_{n}\) are respectively defined as follows:  
\[\hat{\phi}_{n}(\omega)=\left\{\begin{array}{cc}1&\text{if}|\omega|\leq(1- \gamma)\omega_{n}\\ \cos\left[\frac{\pi}{2}\beta\left(\frac{1}{2\gamma\omega_{n}}(|\omega|-(1- \gamma)\omega_{n})\right)\right]&\text{if}\ (1-\gamma)\omega_{n}\leq\mid\omega\mid\leq(1+\gamma) \omega_{n}\\ 0&\text{otherwise}\end{array}\right. \tag{1}\]  
\[\hat{\psi}_{n}=\left\{\begin{array}{cc}1&\text{if}\ (1+\gamma)\omega_{n}\leq\mid \omega\mid\leq(1-\gamma)\omega_{n+1}\\ \cos\left[\frac{\pi}{2}\beta\left(\frac{1}{2\gamma\omega_{n+1}}(|\omega|-(1- \gamma)\omega_{n+1})\right)\right]&\text{if}\ (1-\gamma)\omega_{n+1}\leq\mid \omega\mid\leq(1+\gamma)\omega_{n+1}\\ \sin\left[\frac{\pi}{2}\beta\left(\frac{1}{2\gamma\omega_{n}}(|\omega|-(1- \gamma)\omega_{n})\right)\right]&\text{if}\ (1-\gamma)\omega_{n}\leq\mid \omega\mid\leq(1+\gamma)\omega_{n}\\ 0&\text{otherwise}\end{array}\right. \tag{2}\]  
In Equations (1) and (2), \(\omega\) represents the angular frequency, where the function \(\beta(x)\in C^{k}([0,1])\) is defined as follows:  
\[\beta(x)=x^{4}(35-84+70x^{2}-20x^{3}) \tag{3}\]  
When \(\gamma<\min_{n}\left(\frac{\omega_{n+1}-\omega_{n}}{\omega_{n+1}+\omega_{n}} \right)\), \(\left[\hat{\phi}_{1},\left\{\hat{\psi}_{n}\right\}_{n=1}^{N}\right]\) can satisfy the requirements of a tight frame in the \(L^{2}(R)\) space. An empirical wavelet filter bank is constructed by applying Fourier spectral segment information to Equations (1) and (2), and then the empirical wavelet transform can be defined by using a similar traditional wavelet transform. The detail coefficients are derived from empirical wavelet functions and signals. The inner product is given by:  
\[W_{x}(0,t)=\langle f,\phi_{n}\rangle=\int x(\tau)\overline{\phi_{n}(\tau-t)}d\tau \tag{4}\]  
The detail coefficient \(W_{x}(n,t)\) is defined as the inner product of the detection signal \(f\) and the wavelet scaling function, given by:  
\[W_{x}(n,t)=\langle f,\psi_{n}\rangle=\int x(\tau)\overline{\psi_{n}(\tau-t)}d\tau \tag{5}\]  
The extracted modes \(f_{k}\) is given by:  
\[\begin{array}{l}f_{0}(t)=W_{x}(0,t)*\phi_{1}(t)\\ f_{k}(t)=W_{x}(k,t)*\psi_{k}(t)\end{array} \tag{6}\]  
The detailed algorithm of the EWT can be found in [12].