# Sparse Low-Rank Based Signal Analysis Method for Bearing Fault Feature Extraction
## 4 Simulation Analysis
The simulated signal \(y_{1}\) and the relevant envelope spectrum are displayed in Figure 7. In the waveform, the defect impulses are overwhelmed by heavy noises. In envelope spectrum, the frequencies corresponding to the HCs (marked by red circle) are apparent, but the FCF is overwhelmed by other frequency lines. Therefore, it is difficult to judgment whether the bearing has failed from the raw signal. Next, the simulated signal \(y_{1}\) is processed by FISLRA and TSLRM. Figure 8 represents the results obtained using the FISLRA, in which the extracted repetitive transients are distinct. In the envelope  
Figure 5: Simulated signal \(y_{2}\). Figure 6: Simulated signal \(y_{2}\). (**a**) Analysis result of the traditional sparse low-rank model (TSLRM); (**b**) analysis result of the fault information-based sparse low-rank algorithm (FISLRA). Figure 7: Simulated signal \(y_{1}\). (**a**) Amplitude spectrum; (**b**) envelope spectrum. spectrum, FCF and its harmonics can be recognized clearly. Therefore, there is no doubt that the FISLRA can effectively abstract the repetitive transients from the noisy signal even in the case of interference from HCs. Moreover, to explore the computational efficiency of the proposed FISLRA, a time cost analysis is conducted on an Intel Core i7-9750H desktop computer with a 4.5 GHz CPU. For the simulated signal with a length of 20,000 data points, the average time cost of the proposed method FISLRA is 4 s.  
Then, the same simulated signal \(y_{1}\) is analyzed by using the TSLRM. Figure 9 presents the resultant denoised signal, where the HCs is obvious but no obvious defect impulses can be observed from the amplitude spectrum. And as indicated in Figure 9, the envelope spectrum also fails to provide useful information for fault diagnosis. Moreover, rotating frequency and its harmonics (1\(X\) and 2\(X\)) are very dominant in envelope spectrum, which means that TSLRM is interfered with by the HCs during the extraction of fault features. Therefore, there is no doubt that the FISLRA can effectively abstract the repetitive transients from the noisy signal even in the case of interference from HCs. The superiority of FISLRA over TSLRM can be certified in the section of simulation analysis. Furthermore, to further highlight the effectiveness of the FISLRA, the SK as a widely used and state-of-art method is applied to deal with the same simulated signal \(y_{1}\). Figure 10 displays the kurtogram of the simulated signal, the optimal frequency band with a bandwidth of 312.5 Hz and the center frequency of 6093.75 Hz flagged by a red rectangle is located in the kurtogram. The relevant filtered signal is indicated in Figure 10, from which fault information cannot be found intuitively, and ball pass frequency of outer race (BPFO) and its harmonics are overwhelmed by interference frequencies in envelope spectrum. Therefore, SK fails to extract fault information. The failure result from the assumption that the fault impulses occur in the frequency band with the highest kurtosis. Research has found that kurtosis is more sensitive to a single impulse than to periodic impulses. Therefore, the SK method is susceptible to interference from non-Gaussian signals and random shocks. Figure 8: Fault features extracted by FISLRA. (**a**) Amplitude spectrum; (**b**) envelope spectrum. Figure 9: Fault features extracted by the TSLRM. (**a**) Amplitude spectrum; (**b**) envelope spectrum.