## 4 Conclusions  
In order to decrease the effects of complex interferences on the selection of the IFB, a novel approach, named Mkurtogram, is proposed in this paper, which is applied for fault detection of wheelset bearing. It takes the Mkurt of AC as a new evaluation index to extract fault-induced repetitive transients for the first time, where the impulsiveness and periodicity are taken into consideration, simultaneously. Its main contributions are summarized as follows:  
(i) Apart from improving the SNR and increasing the strength of the periodic peaks, the phase synchronization property of AC process is further revealed in this paper. The AC process ignores the initial phase information, and the signal in the AC domain is synchronous  
Figure 19: Case 3: The results of FK and Autogram. (**a**) the FK; (**b**) the Autogram.  
Figure 18: Case 3: The results of Mkurtogram based on inner race fault period \(T_{i}\). (**a**) the Mkurtogram; (**b**) the filtered signal; (**c**) the SES of (**b**).  
with a phase of 0. As a result, the positions of the fault-related peaks are reordered regularly and the first fault peak will occurs at the time lag of fault period in the AC domain signal, which is beneficial to further distinguish the fault information from interference components.  
(ii) Based on the regular periodic distribution of fault-related peaks in AC domain, a unified representation of the impulsiveness and periodicity is performed by MKurt. Through the directivity of the MKurt, only the reordered fault-related peaks are considered, thus, the influence of background noise and irrelevant periodic interferences is further eliminated.  
(iii) By inputting different fault period, the proposed method can detect the fault-induced repetitive transients with different periods, thus, it can be extended to the compound fault diagnosis of the wheelset bearings.  
The performance of the proposed approach is verified by a simulated signal with multiple interferences and experimental signals collected from two wheelset bearings under different operating conditions. The analysis results show that this method has high anti-interference ability. It can effectively decrease the influence of large impulse interference, the discrete harmonics, and amplitude fluctuation caused by variable working conditions. Further, benefit from periodic directivity of Mkurt, it has successfully applied to the compound faults diagnosis of the wheelset bearing. Compared with FK and Autogram, the results indicate this proposed method is more effectiveness and robustness to extract the wheelset bearing fault feature from complex interferences and to diagnose composite faults.  
Conceptualization: W.L.; data curation: Q.L. and Y.L.; investigation: Q.L. and R.H.; methodology: W.L.; software: S.Y.; validation: X.G.; writing--original draft: W.L.; writing--review and editing: S.Y. All authors have read and agreed to the published version of the manuscript.  
The present research is supported by the National Natural Science Foundation of China (nos. 11790282, 12032017, 11802184, 11902205, and 12002221), S&T Program of Hebei (20310803D), and Natural Science Foundation of Hebei Province (no. A200210028).  
Not applicable for studies not involving human or animals.  
Not applicable for studies not involving human.  
The data are not publicly available due to Laboratory restrictions.  
The authors sincerely thank the reviewers and editors for their valuable comments and suggestions.  
The authors declare no conflict of interest.