# Railway Vehicle Wheel Flat Detection with Multiple Records Using Spectral Kurtosis Analysis
## 3 Methodology for Flat Detection
### Envelope Spectrum Approach to Detect a Wheel with Spectral Kurtosis Analysis  
Envelope spectrum detection is a technique to perform complex demodulation by shifting each frequency to zero and then applying a low-pass filter [59]. Therefore, the signal is multiplied by the following factor to perform a complex signal deformation. \[X(t)=R(t)\exp(2\pi if_{0}t) \tag{6}\]  
where, \(f_{0}\) is the center frequency of the band. After the frequency band with the highest kurtosis level is determined, a passband filter is applied to the raw signal to obtain a higher impulse signal to analyze the envelope spectrum. The process to identify a defective wheel is shown in Figure 6. As shown in this figure, the method to detect a wheel flat is presented with two main blocks. In the first block of the flowchart, the necessary equations for evaluating the envelope spectrum in each sensor of the installation system are explained, and furthermore, the wheel flat is identified in the second block based on the following criteria. If the responses obtained by all sensors are coincident, the amplitude of the envelope spectrum is the same and indicates the passage of a healthy wheel through the system. However, a significant lag in the amplitude of the envelope spectrum infers a passage of a defective wheel. The selection of the demodulation band for the envelope analysis of a defective wheel is frequently performed by comparing the spectrum for a healthy wheel to select the particular frequencies at which the largest change occurred as a result of the fault. It has been identified that spectral kurtosis (\(SK\)) gives a very similar indication of the band to be demodulated without requiring historical data. The spectral kurtosis of a signal is obtained by dividing the main signal into different frequency bands and obtaining the kurtosis of each frequency band [60]. This demonstrates how the peak in the signal changes with frequencies and can be used to identify a frequency band within which a signal has the most impulsive behavior. This information is needed to select the optimal frequency band to demodulate and perform an envelope analysis on the recorded signals. The optimum center frequency and bandwidth combination of a bandpass filter to maximize the kurtosis of the filter output is described by a kurtogram [61]. Figure 6: The methodology to identify a defective wheel. The kurtogram shows the kurtosis results for a range of window lengths and frequencies and is used to visualize the spectral kurtosis. The principle of the proposed algorithm to obtain the kurtogram is based on the multirate filter-bank structure [60; 61]. The algorithm is described first in the case of a binary tree structure and then extended to a 1/3-binary tree structure. The procedure to perform a kurtogram is described in the following steps:  
1. Time-series signal \(R(t)\) is considered, as the response recorded (shear or acceleration) in positions 1 to 19 (presented in Figure 5). 2. Consider \(h\) as a low-pass prototype filter. Then construct two low-pass and high-pass analysis filters \(h_{1}\) and \(h_{2}\), from \(h\), in the frequency bands [0 \(\frac{1}{4}\)] and [\(\frac{1}{4}\) \(\frac{1}{2}\)] of the sampling frequency, respectively, as follow: \[ \begin{array}{l}h_{1}=he^{j\pi/4}\\ h_{2}=he^{j3\pi/4}\end{array} \] (7)  
Filters \(h_{1}\) and \(h_{2}\) are used to perform the low-pass/high-pass decomposition indicated in Figure 7, which is iterated in a pyramidal manner to produce the filter-bank tree, where each level has \(2^{l}\) bands. As illustrated in this figure, let \(R^{i}_{l}(t)\) be a sequence of coefficients issued from the filter \(i\)-th. \(i\) is considered as \(0,1,2,\ldots,2^{l-1}\) at each level (\(l\)) in the decomposition tree. After filtration, the main signal with \(h_{1}\) and \(h_{2}\), kurtosis values would be obtained for each signal. 3.