## 4 Discussion and Conclusions  
This paper proposes an FPO-guided FMLT method to address the smearing problem of TFR arising because the base frequencies of LT do not match the frequencies of signals. The FPO algorithm is first adopted to extract the pre-IF paths of signals, facilitating the construction of bases of FMLT. With the FPO, the proposed method can deal with the signal corrupted with strong noise, where the kurtosis-guided IF estimation may be ineffective. The FMLT with the base frequencies matching with the frequencies of signal components can then be implemented to relieve the smearing problem of TFR  
Figure 14: Inner race fault case: illustration of the constructed frequencies of bases of FMLT. Figure 15: Processing results of the low and resonant frequency band signals of the inner race fault signal using the proposed FMLT: (**a**) FMLT-resulting TFR of the lower frequency band signal, (**b**) FMLT-resulting TFR of the resonant frequency band signal, (**c**) extracted IF ridges from (**a**) using the FPO, and (**d**) extracted IF ridges from (**b**) using the FPO. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**IF Ridges** & **Pre-\(f_{1}\)** & \(f_{1}\)** & **Pre-\(f_{2}\)** & \(f_{2}\)** & **Pre-\(f_{3}\)** & \(f_{3}\)** & **Pre-\(f_{4}\)** & \(f_{4}\) \\ \hline MRE & 0.1839 & 0.0034 & 0.3126 & 0.2941 & 0.1146 & 0.1108 & 0.1632 & 0.0701 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: MREs of extracted IF ridges of inner race fault signal. and then enhance the readability of TFR. More accurate IF paths can be extracted from such enhanced TFR. The MREs of the extracted IF path before (i.e., pre-IF) and after the application of FMLT are calculated, showing that the accuracy of extracted IF paths is improved after the FMLT application. The bearing fault diagnosis can then be finished based on the ratios of arbitrary points on the IF ridges of the low frequency band signal to the ones on the IF ridges of the resonance frequency band signal at the same time instant. Analysis of the results of the experimental signal examines the effectiveness of the proposed method for TFR readability enhancement, accurate IF ridge extraction and bearing health condition monitoring under variable speeds. To summarize, the main advantage of the proposed method is that it solves the smearing problem of TFR by matching the frequencies of bases to the frequencies of signal components. The readability of TFR of signals with variable IF can be improved. As for applications, in addition to bearing fault diagnosis under time-varying speeds, the proposed method can also be applied to gear health monitoring under variable speed operations as well as the TFR enhancement of other signals. It is worth mentioning that the potential limitation of the proposed method is that it may be ineffective in tackling signals with large fluctuations of the rotational speed. This is because the IF ridge in a windowed signal is assumed to be linear; however, if the shaft rotational speed varies with large fluctuations within a short time interval, this assumption may be not true. In this circumstance, a shorter window should be used, which will increase the computational complexity of the proposed method. Conceptualization, J.S. and G.D.; methodology, J.S. ; software, R.D. ; validation, J.S., G.D. and Z.Z. ; writing--original draft preparation, R.D. and J.S. ; writing--review and editing, J.S. and G.D.; supervision, G.D. and Z.Z. ; funding acquisition, J.S. This research was funded by National Natural Science Foundation of China (51605319), Natural Science Foundation of Jiangsu Province (Grant No. BK20160318), Postdoctoral Research Foundation of China (2017M611896) and Natural Science Foundation for Colleges and Universities in Jiangsu Province (Grant No. 16 KJB460020). Authors would like to thank the Lab E026 of University of Ottawa for the data collection and [PERSON] from University of Ottawa for the permission of using the vibration data.