## 4 Conclusions  
This study suggested two signal processing methods to detect wheel flats and diagnose flat sizes. The first one is cepstrum analysis combined with order analysis to consider variable speed conditions. The order analysis converts the time domain into a rotating  
Figure 19: Cross correlation result of the flat wheel using four cycles signal for one flat.  
Figure 20: Normalized amplitude ratios according to flat sizes.  
angular domain. The second one is cross correlation analysis combined with the order analysis for detecting flat signals in severe noise signals.  
The cepstrum analysis is effective in detecting wheel flats. In particular, the cepstrum analysis combined with the order analysis is suitable for application to railway vehicles because speed sensors and tachometers are already installed to measure the rotational speed of the wheels. In order to increase the detectability, the following measures can be summarized: First, use the zero padding to improve the frequency resolution. Second, use the greater amount of sampling data when interpolating in the rotation angle domain; then, it can improve the frequency resolution. In this case, an interpolation using compact data points enough to cover the peak value of the vibration signal must be performed to obtain the equal angular intervals after changing the domains. Otherwise, the sensitivity of the flat signals in the cepstrum analysis is significantly deteriorated.  
As mentioned above, the cepstrum analysis is very useful to detect the periodic signal of wheel flats, but there is a limit to diagnose wheel flat sizes. However, this can be supplemented using the cross correlation analysis. The procedure to diagnose wheel flat sizes using the correlation analysis is as follows: first, flat signals should be changed to the angle domain from the time domain. By doing so, the cross correlation analysis result can be by greatly improved because it is well matched to the kernel function of the wheel flat signal even under variable speed conditions. Second, use the cross correlation kernel function with the uniformity of the wheel flat signal in the angle domain. Several flat signals can be used to amplify the cross correlation flat signals in proportion to its number of each \(2\pi\) angle. The flat signals amplified by the cross correlation analysis will become several times larger than the severe noise signals. Therefore, the advantage of the cross correction analysis is that it can detect flat signals even in severe noise signals. However, the disadvantage is that kernel signals are necessary to apply it. Therefore, to solve for these disadvantages, flat signals should be built in the database when flats occur. If the database is established, the flat size can be diagnosed through the cross correlation analysis. In addition, it is also possible to figure out the flat numbers by counting the amplified flat signals generated during one revolution of the wheel.  
In the near future, the proposed algorithms on the flat signal processing will be provided to railway companies so that they can diagnose the actual faults after inputting them into the equipment using Python code. If it happens, for an actual operation and maintenance, the proposed algorithm can be appropriately applied to measurement, analysis, and diagnosis process of wheel flats in real time.  
Conceptualization, J.S., G.K. and J.K.; methodology J.S., G.K. and J.K.; software J.S. and B.C.; validation J.S., G.K. and J.K.; writing--review and editing, J.S., B.C. and J.K.; supervision, J.K.; project administration, J.K.; funding acquisition, Seoul National University of Science and Technology. All authors have read and agreed to the published version of the manuscript. This study was supported by the Research Program funded by SeoulTech (Seoul National University of Science and Technology). The authors declare no conflict of interest.