## 3 Wheel Flat Recognition and Positioning Algorithm Based on Multisensor Arrays
### Step I: When the Abnormal Impacts Occurred  
The purpose of this step is to determine the specific moments when abnormal impacts occur. The fast Fourier transform (FFT) filter is applied to eliminate the major trends of output signals. With a cut-off frequency of 200 Hz, the high-frequency components caused by wheel flats are extracted, as shown in Figure 17. After the high-pass filtering, the defect signals become much more obvious. Figure 14: Output signals of Plan III under its most unfavorable condition: (**a**) Without wheel flat; (**b**) wheel flat depth of 0.1 mm; (**c**) wheel flat depth of 0.3 mm; and (**d**) wheel flat depth of 0.5 mm. All 10 sensors have sensed two anomalous responses caused by the wheel flat on the front wheel at 0.21 s and 0.34 s.  
Additionally, an interesting fact is that the response amplitudes of sensors to a certain wheel flat seem to obey some special distribution. As the red dashed lines show, the responses to the first impact are almost symmetrically distributed around the location of \(S_{6}\). As for the second impact, the amplitude of response increases along with an increase of the sensor's sequence number. This characteristic provides a hint as to how to locate where the impact happens, which will be later discussed in the Section 3.3. Figure 15: (**a**) The movement of the wheelset and the layout of the multisensor array, the output signals of (**b**) sensor \(S_{4}\); (**c**) sensor \(S_{5}\); and (**d**) sensor \(S_{6}\). Figure 16: Main steps of the wheel flat recognition and positioning algorithm. Figure 17: Processed signals after FFT high-pass filtering. In order to screen out all the impact processes, the sliding window method is introduced in this step. As is shown in Figure 17, the sliding window, with a size of d, moves forward progressively by the step length of \(\delta\). The window is a matrix containing the signals of multiple sensors in a certain time range, which can be expressed as follows:  
\[W_{k}=\left( \begin{array}{cccccc}S_{1},N_{k}&S_{1},N_{k}+1&\ldots&S_{1},N_{k}+ M-2&S_{1},N_{k}+M-1\\ S_{2},N_{k}&S_{2},N_{k}+1&\ldots&S_{2},N_{k}+M-2&S_{2},N_{k}+M-1\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ S_{N_{s}-1},N_{k}&S_{N_{s}-1},N_{k}+1&\ldots&S_{N_{s}-1},N_{k}+M-2&S_{N_{s}-1}, N_{k}+M-1\\ S_{N_{s}},N_{k}&S_{N_{s}},N_{k}+1&\ldots&S_{N_{s}},N_{k}+M-2&S_{N_{s}},N_{k}+M-1 \end{array} \right) \tag{4}\]  
where \(N_{s}\) is the number of sensors. In this case, \(N_{s}\) = 10. \(M\) is the number of samples of each step determined by the window size \(d\) and the sampling frequency \(f\). \(N_{k}\) represents the starting number of the sample in the \(k\)-th step. These two parameters can be calculated using the following equations:  
\[M=d\times f \tag{5}\]  
\[N_{k}=1+(k-1)\times\delta\times f \tag{6}\]  
Outlier tests are performed during every step. Since the filtered signals are not normally distributed, the boxplot method is deployed to distinguish the outliers. This method uses quantiles to determine the thresholds of the normal data, which can effectively eliminate the influence of extreme values on the overall distribution. In light of this, the boxplot method has strong robustness in the outlier test for non-normally distributed data. The upper and lower thresholds of the normal signals of \(i\)-th sensor specified by the boxplot method can be expressed in Equations (7) and (8), respectively [34]. \[Y_{u}=Q_{3}+\alpha\times IQR \tag{7}\]  
\[Y_{l}=Q_{1}-\alpha\times IQR \tag{8}\]  
where \(Q_{3}\) and \(Q_{1}\) are the upper and lower quartiles of the sample, \(IQR\) is difference between \(Q_{3}\) and \(Q_{1}\), and \(\alpha\) is an adjustable coefficient which is often set as 1.5. Outliers are defined as values outside the range between \(Y_{u}\) and \(Y_{l}\). By comparing the maxima and minima of every row of \(W_{k}\) with the calculated thresholds, we discover how many sensors have sensed some abnormal fluctuations in the corresponding time range. Figure 18a illustrates the maxima and minima of the sensor \(S_{1}\) during every sliding step.