# An Improved Empirical Wavelet Transform and Its Applications in Rolling Bearing Fault Diagnosis
## 4 Simulated Signal Verification
### Case Study 3  
The third simulation signal adds a periodic high-frequency impact to a low-frequency cosine signal locally. Periodic high-frequency \(\mathrm{s}_{1}\) impact the signals and simulates faults; low-frequency signals \(\mathrm{s}_{2}\) simulate the normal operation of equipment:  
\[\left\{\begin{array}{c}\mathrm{s}_{1}=2\sin(2\pi 200\mathrm{t})\times(\mathrm{u }(\mathrm{t}-0.09)-\mathrm{u}(\mathrm{t}-0.15)+\mathrm{u}(\mathrm{t}-0.34)- \mathrm{u}(\mathrm{t}-0.4)+\\ \mathrm{u}(\mathrm{t}-0.59)-\mathrm{u}(\mathrm{t}-0.65)+\mathrm{u}(\mathrm{t}- 0.84)-\mathrm{u}(\mathrm{t}-0.9))\\ \mathrm{s}_{2}=10\cos(2\pi 4\mathrm{t})\\ \mathrm{s}=\mathrm{s}_{1}+\mathrm{s}_{2}+\eta\end{array}\right. \tag{21}\]  
where \(\mathrm{u}(\mathrm{t})\) is the step signal, the sampling frequency is 2 kHz, collection time \(\mathrm{t}=1\) s, and \(\eta=\mathrm{SNR}(3)\). The original signal and its constituents are shown in Figure 14. Figure 14a is the original signal that adds noise after superposition; b and c are constituents of the original signal.  
EWT and FEWT were used to process the signals. In the FEWT calculation, the first 60 points of the key function are selected to reconstruct trend components. The threshold \(\lambda=0.0614\). Both methods successfully segment the low-frequency components. The difference lies in the location of the first boundary. The first boundary of the EWT method is closer at 100 Hz, and the FEWT is closer at 10 Hz. A comparison between the two results shows that FEWT is closer to 4 Hz, which results in less noise in the first component of the reconstruction. Figure 15a shows that the spectrum is divided into 14 regions by the EWT method, while FEWT divides the spectrum into only 4 regions. The ineffective regions caused by noise at part B are divided into independent components. This is not accurate. The excessive invalid component not only increases the computational cost and operation time, but also brings a difficulty to the diagnosis.  
In Figure 16, the red line shows the constituents of the original signal; in Figure 16a, the black line is the first 4 of 14 components decomposed by the EWT; Figure 16b is all results of FEWT decomposition. It can be seen from the figure that both the low-frequency and high-frequency components of EWT decomposition contain certain noise. The low-frequency component of FEWT is very smooth. The high-frequency component of FEWT decomposition has a weak low-frequency modulation component but little noise.  
Figure 14: (**a**) The original signal; (**b**) the cosine component; (**c**) the impact component.  
Figure 17 shows the separation success rate and error rate of EWT and FEWT. The success rate of EWT decomposition is 14.29% and FEWT is 50%. FEWT decomposition efficiency and success rate are higher. The cosine component (EM1) and impact component (EM2) are compared, and the results are shown in Figure 17b. It is known that the components decomposed by FEWT are more consistent with the original components. Therefore, the boundary of the proposed method is more reasonable.  
Figure 16: (**a**) The first 4 of 14 components of EWT decomposition modes; (**b**) all components of FEWT decomposition modes.  
Figure 17: (**a**) The success rate of separation; (**b**) the error rate.  
Figure 15: The spectrum decomposition modes of (**a**) the EWT method and (**b**) the FEWT method.