## 4 Validation of the Algorithm  
Validation of the recognition and positioning algorithm was conducted via the experimental method. The offline wheel condition inspection was conducted to measure the radius deviation of the wheels. After several repeats of the experiments, two wheels with relatively severe wheel flats were chosen for the algorithm validation. Figure 20 shows the defect conditions of these two wheels. More specifically, there are two obvious wheel flats with approximate depths of 0.06 mm and 0.04 mm on the front wheel, and one wheel flat with an approximate depth of 0.07 mm on the rear wheel. The measured wheel profiles were input into the simulation model as the front and rear wheels, respectively. The sensor layout is consistent with Section 3. To consider the influence of the monitoring system error, a 5% (SNR = 13 dB) Gaussian noise is added to the calculated output signals. Figure 21 shows the results of the simulation. For easier observation, the baselines of signals are shifted to different degrees. The abnormal data caused by multiple wheel defects mingles together, making it impossible to assess the real conditions of the wheels directly. Figure 21: Raw outputs of sensors during the passage of wheelset under actual service state. Figure 20: The defect conditions of the chosen wheels: (**a**) Front wheel; (**b**) rear wheel. The algorithm proposed in Section 3 was used to identify the defect conditions of the wheels. The curves in Figure 22 illustrate the filtered signals and identified impact processes, while the histogram represents the frequency of outliers during every time step. The results show that there are 8 periods of time identified as potential impact processes. They are denoted as IMP\({}_{1}\) (0.1125-0.11875 s), IMP\({}_{2}\) (0.15625-0.1625 s), IMP\({}_{3}\) (0.18125-0.1875 s), IMP\({}_{4}\) (0.24375-0.25 s), IMP\({}_{5}\) (0.2875-0.29375 s), IMP\({}_{6}\) (0.30625-0.3125 s), IMP\({}_{7}\) (0.38125-0.3875 s), and IMP\({}_{8}\) (0.44375-0.45 s). Furthermore, the position of every impact process, as well as the locations of the wheels, are determined. The results are concluded in Table 2. The initial time for calculating the locations of the front wheel is 0.0595 s. As for the rear wheel, the initial time is 0.17425 s. The reason for the negative values of the rear wheel position is that the rear wheel has not entered the test area at this moment. It is obvious that five abnormal impact processes are generated by the front wheel, and the others are due to the defects of the rear wheel. To better describe the defect conditions of the wheels, the abnormal signals are converted into the polar coordinate system, where the responses represent the deviation of the radius. In this way, the wheel conditions can be deduced and illustrated in Figure 23. The impact signals collected by different sensors converge to the same center angle, reflecting that the wheel flat is most likely to be at this position. From the results of the algorithm, it can be deduced that there are two flat spots on the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Impact** & **Impact** & **Middle Time/s** & **Front Wheel Position/m** & **Rear Wheel Position/m** & **Responsible Wheel** \\ \hline IMP\({}_{1}\) & 0.6–1.2 m & 0.109375 & 1.1225 & \(-\)1.1725 & Front wheel \\ \hline IMP\({}_{2}\) & 1.8–2.4 m & 0.153125 & 1.9975 & \(-\)0.2975 & Front wheel \\ \hline IMP\({}_{3}\) & 0–0.6 m & 0.178125 & 2.4975 & 0.2025 & Rear wheel \\ \hline IMP\({}_{4}\) & 3.6–4.2 m & 0.240625 & 3.7475 & 1.4525 & Front wheel \\ \hline IMP\({}_{5}\) & 4.2–4.8 m & 0.284375 & 4.6225 & 2.3275 & Front wheel \\ \hline IMP\({}_{6}\) & 2.4–3 m & 0.303125 & 4.9975 & 2.7025 & Rear wheel \\ \hline IMP\({}_{7}\) & 6.4–7 m & 0.371875 & 6.4975 & 4.2025 & Front wheel \\ \hline IMP\({}_{8}\) & 5.4–6 m & 0.440625 & 7.7475 & 5.4525 & Rear wheel \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of the recognition and positioning algorithm. Figure 22: The filtered outputs and identified impact processes.