## 4 Cases Studies
### Performance Analysis  
#### 4.2.1 The Proposed Method  
The time domain diagram of the collected original signal is presented in Figure 7. From the signal waveform, it can be concluded that the bearing state experienced three stages: static, linear acceleration and uniform speed. In the linear acceleration phase, the time-domain waveform was stable without obvious periodic impact. It can also be found that the amplitude of the signal was low, and contained some noise interferences. The linear acceleration signal required in this study was obtained from the collected signal. Through the signal waveform, the 40,000 th to 100,000 th points of the acceleration process were selected, and the instantaneous speed at each time point was calculated by the speed pulse signal collected by the tachometer. The equation of the accelerating speed was fitted as follows: \(V=5.0003\times t+6.1888\), where \(V\) is the instantaneous speed (Hz) and \(t\) is the sampling time point (s). Figure 5: The test rig and installation locations of the microphone sensor. Figure 6: The fault bearing, with defects in (left) inner and our race, (right) rolling element. In this study, the variation of bearing speed was slow and linear, so an angle domain resampling method was used to process the variable speed signal. The angular resampling frequency was set as 200 Hz, i.e., 200 points were sampled per revolution. The angular domain waveform and envelope order spectrum of the original signal are respectively shown in Figures 7 and 8 as determined by the fitting of the speed equation. The fault characteristic orders of the bearing in the angle domain are listed in Table 1.  
Because the original signal-to-noise ratio (SNR) of the variable speed signal was very low, as shown in Figure 9, the extraction of the fault characteristic order after angular domain resampling was difficult. It can be determined from the envelope order spectrum that the amplitude of the signal was very small. There was only a low-frequency component in the low-frequency part, which does not accurately reflect the state of the bearing. The proposed method was then used to extract the bearing fault characteristic order under such complicated conditions, and the superiority of the proposed method was proven. In the first step, the DEWT-SVD was used to estimate the source number. After the signal was decomposed by DEWT, 10 components were obtained. These 10 components were combined with the original signal to form a virtual multi-channel observation signal. The virtual observation signal was then processed by SVD, and 11 singular values were obtained, as presented in Table 2. The adjacent singular value difference is presented in Table 3. Figure 10 illustrates the broken line diagram of the adjacent singular value difference. It is evident from Tables 2 and 3 and Figure 10 that the singular value change after \(\lambda\)4 was very small, as was the difference between \(\lambda\)4 and \(\lambda\)5, which was about equal to 0. Hence, it can be concluded that the number of source signals of the compo \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Outer-Race (\(f_{o}\))** & **Roller (\(f_{p}\))** & **Inner-Race (\(f_{p}\))** \\ \hline Order & 4.05 & 5.05 & 5.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the bearing. Figure 8: The angular domain waveform of the original signal. Figure 7: The time domain waveform of the collected original signal. is consistent with the three fault sources of the inner ring, outer ring, and rolling element fault that had been artificially processed prior. The results of this step indicate that the proposed DEWT-SVD source number estimation method exhibits superior performance. It can be clearly seen from Figure 10 that the value of Number 4 is close to 0, which is also an obvious inflection point. At the same time, due to the low signal-to-noise of the bearing acoustic signal, there is a large noise interference in the low-frequency region, so set \(N=4\).