## 3 Application and Results
### Fault Diagnosis of Bearing with Fault at the Outer Raceway  
The time-domain waveform of the fault signal \(x_{out}\) from the outer raceway is shown in Figure 9a, where the sampling number is 12000. Divide the signal \(x_{out}\) into four segments of \(x_{out1}\), \(x_{out2}\), \(x_{out3}\), and \(x_{out4}\). The number of each segment is 2048. Figure 9b displays the time-domain waveform of \(x_{out1}\). LMD is used to decompose \(x_{out}\) and \(x_{out1}\) separately into a series of PFs and \(u_{m}(t)\) is the residual component, as shown in Figure 10. The correlation coefficients \(\rho\) between the raw signal and its PFs were calculated, as shown in Table 3. Similarly, it is obvious that both of \(PF_{1}(t)\) contain most fault information. Figure 8: Comparison chart of the Hilbert envelope spectrum of \(PF_{1}(t)\) after calculating the remaining three segments of data: (**a**) Without spectrum correction; (**b**) with spectrum correction. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\rho\) & \(PF_{1}(t)\) & \(PF_{2}(t)\) & \(PF_{3}(t)\) & \(PF_{4}(t)\) & \(PF_{5}(t)\) & \(PF_{6}(t)\) \\ \hline \(\rho_{out}\) & 0.9986 & 0.0602 & 0.0106 & 0.0007 & 0.0001 & 3.605E\(-\)5 \\ \(\rho_{out1}\) & 0.9992 & 0.0380 & 0.0090 & 0.0005 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The correlation coefficients \(\rho\): \(\rho_{out}\) is the correlation coefficients between \(x_{out}\) and its PFs; \(\rho_{out1}\) is the correlation coefficients between \(x_{out1}\) and its PFs. Figure 9: Time-domain waveforms of \(x_{out}\) and \(x_{out1}\) with different sampling number: (**a**) The sampling number of \(x_{out}\) is 12,000; (**b**) the number of first segment \(x_{out1}\) is 2048. The comparison chart of the Hilbert envelope spectrum of \(PF_{1}(t)\) which is from Figure 10a is shown in Figure 11. The fault frequency 103.3 Hz, which is not corrected, is basically equal to the theoretical fault frequency 103.1 Hz, as shown in Figure 11a, and the double frequency 207.3 Hz exists obviously. Similarly, the fault frequency 103.5 Hz, which is corrected, is almost equal to the theoretical fault frequency 103.1 Hz, as shown in Figure 11b, and the double frequency 207 Hz exists also obviously. Whether the fault frequency is corrected or not, it is almost equal to the theoretical fault frequency, but the corrected fault frequency is farther from the theoretical fault frequency than the uncorrected fault frequency. The reason is that the theoretical solution to the ratio correction method is affected by the noise. When the SNR is low to a certain extent, the spectral line of the spectrum will be overwhelmed by the noise, which may cause an initial positioning error on the signal [35]. Without the noise between the frequencies, the algorithm of the ratio correction method is simple and its calculation accuracy is high. However, the key to the correction accuracy is to find the most maximum lines of \(y_{k}\) and \(y_{k+1}\) accurately in the main lobe, as shown in Figure 1b, and then get the correction frequency according to the Equation (6). If the SNR is relatively high, the two maximum lines in the main lobe can always be found correctly. When SNR is low to some extent, the amplitude of the maximum spectral line becomes small. It is easy to find a sub-largest line in the opposite direction and then result in a wrong line correction, which will reduce the accuracy of the correction [36]. Due to the interference of the noise or frequency leakage, even in the case of high SNR, there are some cases that the true spectral line may be wrongly selected [24; 37]. In this regard, a more in-depth research is needed. However, when analyzing the long-time signals, it does not affect the accuracy of the fault frequency whether the fault frequency is corrected or not. Figure 10: PFs of \(x_{out}\) and \(x_{out1}\) with different sampling number: **(a)** The sampling number of \(x_{out}\) is 12,000; **(b)** the number of first segment \(x_{out1}\) is 2048.