# An Improved Empirical Wavelet Transform and Its Applications in Rolling Bearing Fault Diagnosis
## 4 Simulated Signal Verification
### Case Study 2  
The second signal simulates the working environment when the equipment breaks down. First, the periodic shock of equipment failure is simulated: a set of damped impulse signals are established and the inherent frequency of the impulse signals f\({}_{\text{n1}}=1000\) Hz, the damping coefficient g = 0.007, and the repeated cycle T = 0.01 s. So the characteristic frequency f\({}_{\text{q}}=100\) Hz. Second, the nonimpact and nonfault modulation signals s\({}_{\text{2}}\) in rotating machinery operation are simulated. The inherent  
Figure 8: The empirical modes (EMs) of (**a**) EWT method and (**b**) FEWT method. Figure 9: (**a**) The success rate of separation; (**b**) error rate. frequency of this amplitude-modulated and frequency-modulated (AM-FM) signal is \(\rm{f_{n1}}=3000\) Hz, the width of the side band is 100 Hz, and the noise is \(\rm{\eta}=SNR(-0.03\) db). \[\left\{ \begin{array}{c}\rm{s_{1}}=7e^{-g\times 2\pi f_{n1}t}\times\sin \Bigl{(}2\pi f_{n1}t\times\sqrt{1-g^{2}}\Bigr{)}\\ \rm{s_{2}}=2\sin(100\pi t)\sin(2\pi f_{n2}t+\sin(200\pi t))+0.8\cos(4\pi t)\cos (100\pi t)\\ s=\rm{s_{1}}+\rm{s_{2}}+\rm{\eta}\end{array} \right. \tag{20}\]  
Figure 10a,b shows the components of the signal. The signal-to-noise ratio signal is shown in Figure 10c and its spectrum in Figure 10d. The central frequencies of the two components are far apart in the spectrum. Theoretically, EWT can distinguish these two components, but the difficulty in dividing boundaries of this kind of signal is reducing the noise interference and invalid components. In this paper, two methods are used to process the signal and the following results can be obtained. Figure 11a is the result of EWT. The impulse signal with an inherent frequency of 1000 Hz and the modulation signal with an inherent frequency of 3000 Hz are successfully separated. However, the following problems are also left: part A in Figure 11a has a lot of noise left; part B divides up too many invalid components. Figure 11: The boundaries divided by (**a**) the EWT method and (**b**) the FEWT method (green line: trend component). Figure 10: (**a**) The pure signal; (**b**) the amplitude-modulated and frequency-modulated (AM-FM) signal; (**c**) the signal-to-noise ratio (SNR) signal; (**d**) the spectrum. Figure 11b shows the result of the FEWT method. The first 30 points of the key function are selected to reconstruct the trend component. The threshold \(\lambda=0.0833\). After calculation, FEWT divides the frequency domain into seven parts, five fewer than EWT. Part A, which contains more noise in the result of EWT, is divided by the FEWT method. The FEWT method obtains fewer components than EWT. The reduction is shown in part B. Since part B is mostly noise, FEWT is more reasonable than EWT. Two parts representing pulse information and modulation information are extracted: for the result of EWT, the third and eighth components are extracted; for the result of FEWT, the fourth and sixth components are extracted, as shown in Figure 12. Figure 12a,b show pulse and modulation information obtained by the EWT method, and Figure 12c,d by FEWT. Because the boundaries of the two methods are similar, the reconstructed components are almost the same, but the details are different. The modulation signal decomposed by EWT contains a lot of noise, which is segmented by EWT. Enlarging the data between 0.04 s and 0.06 s at part A in Figure 12b,d, such as Figure 12e,f, abnormal information can be found in the details of the modulated signal decomposed by EWT, as shown in part B. A similar signal is also found at the corresponding position C of the modulated signal decomposed by FEWT, but the amplitude is relatively small. It can be seen that FEWT can not only divide fewer and more reasonable modal components than EWT, but also reduce certain noise effects when processing pulse signals with modulated signals. The simulation results show that the FEWT method is superior to the EWT method. Figure 13 shows the separation success rate and error rate of EWT and FEWT.