# Train Wheel Condition Monitoring via Cepstral Analysis of Axle Box Accelerations
## 5 Discussion
### Navewumber Analysis  
The navewumber analysis provides a robust tool to extract the wheel circumference from the ABA data. It can be recovered with a resolution equal to the distance between two measurement points, which depends on the train speed and sampling frequency. At a speed of 20 m/s and sampling frequency of 20 kHz the cepstral resolution is 0.001. We could show that between 80 and 90 percent of the calculated wheel circumferences were within a range of two millimeters along the whole train journey.  
The high resolution and accuracy allow precise monitoring of the wheel diameter and therefore enable wheel-wear trend analysis. The approach provides reliable results under varying operational conditions. We found out that a minimal train speed of 15 m/s is sufficient to allow the estimation of the wheel circumference. Above this speed, the navewumber analysis is not affected by the train speed but its accuracy directly depends on the accuracy of the speed estimation that is used to transform the data from the time to distance domain. Thus, short-term variations of the estimated wheel diameter most likely relate to speed estimation errors. These Gaussian distributed errors can be estimated by the positioning algorithms introduced in Section 4.2.  
It is especially noteworthy that the track condition only has a minor effect on the estimation of the wheel circumference. Even at track segments, where high wheel vibrations were excited, the wheel circumference could be determined accurately.  
Due to the conicity of the wheels, the gauge and curve radius might have an influence on the estimated wheel circumference. However, the data analyzed in this study were not affected by the track geometry. Furthermore, the wear-related reduction of the wheel diameter is monotonic, while track-geometry changes only occur at certain track segments. Therefore, trend analysis of the wheel wear is not influenced by the track geometry. The influence of the track conditions can be further reduced by averaging over several track segments or taking longer track segments into account.  
The synthetic data analysis has shown that the absolute amplitude of the cepstral peak can be similarly influenced by the track as well as wheel conditions. A mildly defected wheel running on a track with low roughness can produce a peak similar to that produced by a more severely defected wheel on a track with higher roughness. More generally, the amplitudes of the peaks in the navewumber domain provide a measure of the repeatability of the ABA signal and hence a measure of the relative contribution of the wheel condition to the dynamic wheel rail interaction.  
The discrimination of different wheel defects using cepstral peak position and amplitude alone seems to be impracticable. Nevertheless, the cepstral features might complement other data driven approaches for wheel defect diagnosis.  
It should be noticed that the measurement campaign, which builds the data base for this study did not focus on wheel monitoring but rather on testing the performance of the multi-sensor system. Therefore, no ground truth on the actual condition of the wheel exists.  
Nevertheless, no severe wheel defects were observed during operation. Hence, further tests including ground truth measurements and calibration by means of direct wheel profile measurements are required to determine thresholds for wheel defect detection. In principal, the methodology provided here, could be similarly used for the on-board detection of bearing defects, which should be subject of future studies.