## 3 Fault Feature Extraction and Enhancement of Rolling Element Bearing Based on MCKD-EWT  
### Improved Empirical Wavelet Transform  
The EWT provides an adaptive decomposition of the signal by detecting local maxima or minima of the spectrum; a large number of pseudo-maxima and pseudo-minima caused by the strong background noise will lead to an improper spectrum segmentation. Some researchers [34; 35] have proposed modifications about the spectrum segmentation of the EWT. However, these modifications ignore the spectrum shape and fail to get a good segmentation in \(-5\) dB or higher noise conditions. In order to avoid the inaccurate identification of these modes caused by the heavy background noise, an improved EWT method is proposed in this paper. A more reliable segmentation methodology for the signal spectrum is introduced by calculating the envelope curve of the amplitude spectrum of the signal. We calculate the envelope curve at first, and then modify the envelope curve with a pre-set threshold according to the SNR value. Finally, we segment the signal spectrum based on the local extrema of the modified envelope curve. The detailed progress of improved EWT is listed as the following:  
1. Conduct FFT and obtain the Fourier spectrum \(X(\omega)\) of the signal \(x(t)\). 2. Calculate the following threshold according to the method of literature [35]. \[\lambda=A_{l}+r(A_{h}-A_{l})\] (19) \[r=C/SNR\] (20) where \(\lambda\) denotes the pre-set threshold, \(A_{h}\) and \(A_{l}\) are, respectively, the maximum and minimum magnitudes of the spectrum and \(r\) is inversely proportional to the SNR. Herein, a constant \(C\) is used to control the level of threshold. The larger the value of \(C\), the higher the level of threshold tends to be. In this paper, \(r\) is taken to be 0.1 based on the assumption that the SNR of signals is higher than 85 dB and \(C=10\). The steps are continued as follows:
3. Calculate the envelope curve of the amplitude spectrum based on the local maxima and minima by linear interpolation method, then modify the envelope curve according to the threshold, and finally detect all the extras of the modified envelope curve. If the number of the local maxima points above the threshold is larger than the pre-defined number \(N\) of the components, then keep on calculating the envelope curve of the spectrum until the number of the local maxima points is equal or less than the pre-defined number \(N\) of the components. The number for the iteration process of the calculating envelope is always 5. 4. Obtain the segmentation boundaries. Detect all the local extrema points of the modified envelope curve, and then get the frequency bands of EWT modes by choosing the frequencies of local maxima points as the central frequencies of the modes. After that, locate the boundaries of each mode. Differing from the segmentation method, which uses the local minima points of the modified envelope curve as the boundaries of the EWT modes introduced in the literature, this method chooses the midpoints of the adjacent local minimum points as the segmentation boundaries. The detailed steps firstly calculate the midpoints of the adjacent local minimum points, and then sort them in ascending order, i.e., \(\omega=[\omega_{1},\cdots,\omega_{n-1},\omega_{n},\cdots,\omega_{N}]\), which are used to segment the frequency spectrum. Finally, choose the \(\omega_{n}\) between the adjacent local maximum points as the boundaries. 5. Segment the spectrum and choose the most meaningful component. Calculate the kurtosis value of each component and choose the component with the maximum kurtosis value to further detect the fault feature. In the improved EWT method, the interference frequencies induced by the background noise can be restrained in the envelope curve of the signal spectrum. Therefore, the segmentation can be more accurate. Additionally, the numbers of the modes can be readily determined according to the detailed envelope shape of the actual signal.