## 4 Experimental Verification
### Analysis Result of the Outer Race Fault Signal
The second is Figure 7a, whose noise removal is not complete, although the vibration of the fault signal is embodied especially at 0-0.06 s. The waveforms in Figures 7c and 7d have obvious differences compared with the former two and fail to reflect the oscillation of the fault signal. For the better identification of the de-noised signal, the reconstructed signals are demodulated with the Hilbert transform, and the envelope spectrum at each band is obtained. Figure 6: Time-domain waveform of the early out race fault in bearing 1. Figure 7: Time-domain waveform of the de-noised early outer race faults in bearing 1: (**a**) De-noised signal by MMBDSOICE; (**b**) De-noised signal by MMBDSOICE; (**c**) De-noised signal based on MPDSOISE; (**d**) De-noised signal based on MPOCSOISE. It can be seen from Table 2 that the characteristic frequency of the outer race fault _f_O is 236.4 Hz. Figure 8a shows that the whole envelope spectrum is covered by the noise in the early outer race fault, and the characteristic frequency is labeled \"out\". As a comparison of the rest of the figures, characteristic components of faulty signals are lost in Figure 8b,c. In addition, there is an irrelevant component whose frequency is 48.39 Hz in Figure 8b, as well as 48.38 and 3964 Hz in Figure 8c. However, these frequencies are not inherent but are due to the periodic defects of bearings in the process of production [25], and this will cause the low reliability of the fault diagnosis resulting from these disturbances. All of the characteristic frequencies are reconstructed in Figure 8d, although some other invalid components have been introduced. It can be inferred that the effectiveness of the de-noised signal based on MMBCSOICE is better than others. It can be seen in Figure 9 that the apparent transient impacts arose in the spectrum because of the serious outer race defect, which was buried by strong noise. Figure 10 exhibited the time domain spectrum of the de-noised signal based on different methods. The vibration of the fault signal is reconstructed integrally, and the transient signal can be obtained through applying the proposed method in Figure 10b. Nevertheless, the transient signals extracted by other methods such as MPOCSOISE and MPODSOISE still have strong noise. Figure 11 shows the filtered signal and its envelope spectrum. Mechanical equipment with gears and rolling bearing faults generally has a periodic pulse impact and exhibits the phenomenon of modulation of the vibration signal, which shows a uniform spacing modulating sideband of the meshing frequency or natural frequencies in the spectrum. In the first picture Figure 11a, the spectrum is so polluted by the noise that this paper can only find the characteristic frequency, _f_O, and its two harmonics (e.g., _f_O, _2_f_O, _3_f_O). The most harmonics preserved in the envelope spectrum of the de-noised signal are in Figure 11d. It can be seen from the spectrum characteristic frequency that six harmonics are restored (e.g., _f_O-_7_f_O). On the contrary, the component of characteristic frequency was not recovered by MPODSOISE and MPOCSOISE in Figure 11b, and only the characteristic frequency _f_O can be found in the envelope spectrum in Figure 11c. Figure 8: The envelope spectrum of the early outer race faults in bearing 1: (**a**) Original faulty signal; (**b**) De-noised faulty signal based on MPODSOISE and MPOCSOISE; (**c**) De-noised faulty signal by MMBDSOICE; (**d**) De-noised faulty signal by MMBCSOICE. In the process of extracting bearing outer race faults based on SVD, MMBCSOICE succeeded in locating the position of demarcation at which the singular value of the useful signal and the noise became separated. It is known from the experimental analysis that the number of effective singular values used to be small based on MPODSOISE and MPOCSOISE, which may cause the loss of important information about useful signals. It can be seen from Figures 8 and 11 that there is no characteristic frequency in the envelope spectrum.