# An Optimal Resonant Frequency Band Feature Extraction Method Based on Empirical Wavelet Transform
## 3 Optimal Resonant Frequency Band is Established Based on the EWT
### Average Spectral Negentropy Operator  
Spectral kurtosis, as a statistical tool, can be used to detect the anomaly of the system by its sensitivity to singular signals. It is defined in the frequency band according to the spectral cumulant as follows:  
\[K_{x}=\frac{S_{4x}(IMF_{i})}{S_{2x}^{2}(IMF_{i})}-2=\frac{\left\langle\left| SE_{x}(n;IMF_{i})^{4}\right|\right\rangle}{\left\langle\left|SE_{x}(n;IMF_{i})^{2 2}\right|\right\rangle}-2 \tag{9}\]  
where \(S_{4x}(IMF_{i})\) is the fourth-order spectral cumulant, \(S_{2x}^{2}(IMF_{i})\) is the \(2n\)-order instantaneous moment. \(\langle\cdot\rangle\) is the mean operation, and the discrete signal \(x(n)\) of length \(n\) is defined, whose squared envelope (SE) on the \(i\)-th mode is defined as follows:  
\[SE_{x}(n;IMF_{i})=\left|x(n;IMF_{i})+jH((n;IMF_{i}))\right|^{2} \tag{10}\]  
where \(H(\cdot)\) is the Hilbert transform. Fault pulse in a mechanical system causes a sudden change in the vibration signal, it means that the entropy value of the system also changes. When the bearing is in a normal state, the energy fluctuation of the signal is constant and the spectral entropy value reaches a maximum. Conversely, when the fault pulse causes the energy fluctuation to change, the spectral entropy is at a minimum, which is the opposite of the kurtosis index. To maintain the same physical significance as the spectral kurtosis for detecting the transient change of the system, the Spectral Negentropy of the time domain was taken to characterize the pulse characteristics on all modes, defined as follows:  
\[\Delta I_{e}(IMF_{i})=\left\langle\frac{SE_{x}(n;IMF_{i})^{2}}{\left\langle SE_{ x}(n;IMF_{i})^{2}\right\rangle}\ln(\frac{SE_{x}(n;IMF_{i})^{2}}{\left\langle SE_{ x}(n;IMF_{i})^{2}\right\rangle})\right\rangle \tag{11}\]  
As can be seen, the Spectral Negentropy in the time domain can be regarded as the spectral kurtosis with a weight of \(\ln(SE_{x}(n;IMF_{i})^{2}/\left\langle SE_{x}(n;IMF_{i})^{2}\right\rangle)\). However, due to the influence of noise, only the pulse characteristics will have large errors, the cyclostationarity characteristics shown when a local faults exist should also be taken into account at the same time. Periodic pulse faults appear as a series of repeated transients in the time domain (i.e., harmonics in the square envelope spectrum). Therefore, the spectral negative entropy in the frequency domain is used to reflect the cyclostationary characteristics of mode \(i\), defined as follows:  
\[\Delta I_{E}(IMF_{i})=\left\langle\frac{SES_{x}(\alpha;IMF_{i})^{2}}{\left\langle SES _{x}(\alpha;IMF_{i})^{2}\right\rangle}\ln(\frac{SES_{x}(\alpha;IMF_{i})^{2}}{ \left\langle SES_{x}(\alpha;IMF_{i})^{2}\right\rangle})\right\rangle \tag{12}\]  
In the formula: \(SES_{x}(\alpha;IMF_{i})\) is the square envelope spectrum (SES) on the \(i\)-th mode, defined as:  
\[SES_{x}(\alpha;IMF_{i})=F(SE_{x}(n;IMF_{i}) \tag{13}\]  
In this formula \(F\) is the Fourier transform and \(\alpha\) is the frequency change value. When the transient fault pulse exists in the bearing, both the values of \(\Delta I_{e}(IMF_{i})\) and \(\Delta I_{E}(IMF_{i})\) increase. Combining them can simultaneously measure both impulsiveness and cyclostationarity. Then, the expression of the average spectral negentropy (ASN) of the \(i\)-th mode is as follows:  
\[\Delta I_{1/2}(IMF_{i})=\frac{\Delta I_{e}(IMF_{i})}{2}+\frac{\Delta I_{E}(IMF _{i})}{2} \tag{14}\]