# An Improved Empirical Wavelet Transform and Its Applications in Rolling Bearing Fault Diagnosis
## 4 Simulated Signal Verification
### Case Study 1  
The first simulation signal consists of three cosine signals with different frequencies and amplitudes. \[\left\{ \begin{array}{l}\mathrm{s}_{1}=2.0\cos(2\pi\mathrm{f}_{1}\mathrm{t}) \\ \mathrm{s}_{2}=1.5\cos(2\pi\mathrm{f}_{2}\mathrm{t})\\ \mathrm{s}_{3}=2.5\cos(2\pi\mathrm{f}_{3}\mathrm{t})\\ s=\mathrm{s}_{1}+\mathrm{s}_{2}+\mathrm{s}_{3}+\eta\end{array} \right. \tag{17}\]  
Among them, \(\mathrm{f}_{1}=100\) Hz; \(\mathrm{f}_{2}=150\) Hz; \(\mathrm{f}_{3}=300\) Hz; and \(\eta=\mathrm{SNR}(3)\). The composition, noise, and spectrum of the simulation signal are shown in Figure 6. The signal contains three frequency components: 100 Hz, 150 Hz, and 300 Hz. The frequencies of the first two cosine signals are close to each other. How to successfully extract these three components and minimize the noise interference in the extracted components becomes an important problem to be solved. Figure 5: The traditional threshold functions. First, the proposed EWT and FEWT methods are used to process the signals. Figure 7 shows the effect comparison of the two methods in terms of boundary segmentation. The long red dashed lines are the boundaries of FEWT and the short blue dashed lines are the boundaries of EWT. It can be seen from the figure that the blue boundaries divide the spectrum into too many areas. Although the components represented by f\({}_{3}\) were successfully extracted, f\({}_{1}\) and f\({}_{2}\) were not distinguished. This is one of the biggest defects of EWT. Another major defect, dividing too many regions, will lead to the same amount of invalid components in the next reconstruction, which may increase the workload of the subsequent analysis and interfere with diagnosis. In the process of signal processing by the FEWT method, the first 180 points are selected from key functions to reconstruct trend components. After calculation, the soft threshold denoising process is \(\lambda=0.2309\). The minimum point in the trend component after noise reduction can be well separated into three frequencies and contains only a few unavoidable noises. In Figure 7, the regions divided by long red dashed lines are more reasonable than the short blue dotted lines. Since EWT decomposes too many components, only the first three components of the EWT containing all three simulation signals are extracted for comparison. As shown in Figure 8a, the EWT method does not separate the components of 100 Hz, 150 Hz, and 300 Hz. There is modal aliasing in EM1, and EM2 is the component with a frequency of 300 Hz. However, because it contains more noise, it causes a very unstable modulation phenomenon. EM3 and the remaining components are invalid components that only contain noise. In the process of processing the signal by FEWT, under the effect of threshold noise reduction, the spectral boundary shrinks closer to the frequency of the component. The components representing the 2, 4, and 6 frequency domains are shown in Figure 8b. It can be seen from the figure that the three components are perfectly separated. The amplitudes of the three components have only a small fluctuation, which proves that the noise contained in the signal is small. It is proved that the proposed method is suitable for signals with relatively close frequencies. At this point, the FEWT method is superior to the EWT method. Figure 6: (**a**–**c**) The constituent components of (**d**); (**e**) signal-to-noise signal; (**f**) spectrum of (**e**). Figure 7: The spectrum of the signal (black line); boundaries detected by the EWT method (blue lines); boundaries detected by FEWT method (red lines). For the simulation signal, this paper verifies the rationality of the FEWT boundary from two aspects: the success rate of separation and the error rate. The formula for the success rate of separation is as follows:  
\[\text{rate\_Success}=\text{effective(EM)}/\text{all(EM)} \tag{18}\]  
where rate_Success is the success rate of separation, effective(EM) is the number of empirical modes successfully separated, and all(EM) is the total number of empirical modes.