## 4 Experimental Validation and Discussion
### STFA-PD of Vibration Signals of Rolling Bearings  
Figure 6 shows the time-domain waveforms of the vibration signals of bearingss with different fault severity degrees at the inner race. Comparing Figure 7a with Figure 7b-d, it is obvious that we can easily determine whether a fault occurs or not, as the signal of the inner race with a fault of 0.000 in was a normal signal distinct from fault signals. From Figure 7b-d, we find that all fault signals have prominent impact components with different amplitudes. However, the differences between the signals of different fault severity degrees were insufficient to accurately distinguish the fault severity degrees of the inner race. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Test** & **Fault Type** & **Severity Degree (inches)** \\ \hline \multirow{3}{*}{1} & \multirow{3}{*}{Inner race fault} & 0.000 \\  & & 0.007 \\  & & 0.014 \\  & & 0.021 \\ \hline \multirow{3}{*}{2} & \multirow{3}{*}{Outer race fault} & 0.000 \\  & & 0.007 \\  & & 0.014 \\  & & 0.021 \\ \hline \multirow{3}{*}{3} & \multirow{3}{*}{Ball fault} & 0.000 \\  & & 0.007 \\  & & 0.014 \\ \cline{1-1}  & & 0.021 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test arrangement of the bearing fault severity. Figure 7: Time-domain waveforms of the vibration signals of bearings with four different faults severity degrees at the inner race: (**a**) 0.000 in, (**b**) 0.007 in, (**c**) 0.014 in, and (**d**) 0.021 in. In this study, the STFA-PD method was employed to process the vibration signals. For comparison, the STFT, PWVD, and WT were used to conduct the time-frequency analysis of the signals with different fault severity degrees at the inner race. Figures 8-11 show the TFIs obtained using the STFT, PWVD, WT, and STFA-PD, respectively. The energy distribution in the frequency domain of the normal signal was found to be narrow and relatively centralized in the low-frequency band. When an inner race fault occurs, the energy distributions became very wide, and the energy was mostly concentrated in the high-frequency band ranging from 2500 Hz to 3500 Hz. All TFIs obtained from the fault signals showed certain proportions of high-frequency impact components corresponding to the time-domain waveforms of the vibration signals. Moreover, the proportion of high-frequency impact components increased with an increase in the fault severity degree. Figure 8 shows that the time-frequency concentration and resolution of the traditional STFT were low and the TFIs could not accurately express the time-frequency characteristics of the vibration signals. In addition, there were obvious noise interferences in the TFIs. On the contrary, the TFIs obtained using the PWVD were good in terms of the time-frequency concentration and resolution, and the TFIs of different fault severity degrees could be accurately distinguished. However, they suffered from cross-term interferences, and thus, could not precisely reflect the time-frequency characteristics of the vibration signals, as shown in Figure 9. As shown in Figure 10, similar to the STFT method, the TFIs obtained using the WT were not subjected to cross-term interferences. Moreover, we found that the WT had a relatively good distinctiveness. However, the frequency resolution obtained using the WT was lower than that obtained using the STFT, although its time resolution was better. Additionally, as the WT is essentially a type of time-scale transform and there is no good correspondence between scale and frequency, the WT could not accurately express the frequency changes in the signals with respect to time. Figure 8: TFIs obtained using the STFT from bearing vibration signals of different severity degrees: (**a**) 0.000 in, (**b**) 0.007 in, (**c**) 0.014 in, and (**d**) 0.021 in. Figure 10: TFIs obtained using the WT from bearing vibration signals of different severity degrees: (**a**) 0.000 in, (**b**) 0.007 in, (**c**) 0.014 in, and (**d**) 0.021 in.