## 2 Cepstrum Analysis for Wheel Flat Detection  
### Process of the Cepstrum Analysis for Harmonic Signal Detection  
In 1963, [PERSON] et al. [13] proposed a cepstrum signal processing method that can classify and express the first signal and echo signal by the inverse Fourier transformation of logarithm of spectrum signals. After that, several papers [14; 15] used it as a signal processing method to detect faults of rotational machines. The cepstrum analysis is effective in detecting periodic components of spectrum signals. It performs Fourier transform, logarithmic calculation, and Fourier inverse transform in order for measured time domain signals [16]. If there is a signal for two wheel flats with time delays \(t_{1}\) and \(t_{2}\), the signal \(x(t)\) can be represented by a convolution of one flat signal with two impulses, as shown in Equation (1):  
\[ \begin{array}{l}x(t)=v(t-t_{1})+\alpha v(t-t_{2})\\ =v(t)*[\delta(t-t_{1})+\alpha\delta(t-t_{2})]\end{array}  \tag{1}\]  
where \(\alpha\) is a proportional ratio of the second flat signal magnitude to the first one. Using Fourier transform of \(x(t)\), the convolution of the time domain is transformed into an equation of the frequency domain, as shown in Equation (2). It is represented by Euler formula as both real values of the cosine function and imaginary values of the sine function with \(1/t_{1}\) and \(1/t_{2}\) cycles in the frequency domain:  
\[ \begin{array}{l}X(f){=}\int_{-\infty}^{\infty}x(t)e^{-jwt}dt=V(f)\big{[}e^{ -iwt_{1}}+\alpha e^{-iwt_{2}}\big{]}\\ =V(f)\big{[}(cos\omega t_{1}+cos\omega t_{2})-i(sin\omega t_{1}+sin\omega t_{2} )]\end{array}  \tag{2}\]  
The frequency magnitude after Fourier transform is calculated as follows:  
\[ \begin{array}{l}|X(f)|=|V(f)|\sqrt{1+\alpha^{2}+2\alpha(cos\omega t_{1}cos \omega t_{2}+sin\omega t_{1}sin\omega t_{2})}\\ =|V(f)|\sqrt{1+\alpha^{2}+2\alpha cos2\pi f(t_{2}-t_{1})}\end{array}  \tag{3}\]where \(\omega=2\pi f\). The signal for two flats with time delays of \(t_{1}\) and \(t_{2}\) becomes a spectrum having a period of \(1/(t_{2}-t_{1})\) in the frequency domain. Next, after taking the natural logarithm of its squared value, it is divided into the flat signal component and the impulse-periodic component as shown in Equation (4). Here, by performing Fourier inverse transform, the cepstrum can be obtained as Equation (5), which is composed of a summation of a flat signal \(C_{V}(\tau)\) and \(C_{I}(\tau)\) impulse. The horizontal axis \(\tau\) of the cepstrum has the same dimension to the time t, and is defined as Quefrency:  
\[\ln|X(f)|^{2}=\ln|V(f)|^{2}+\ln\Bigl{[}1+\alpha^{2}+2\alpha cos2\pi f(t_{2}-t_ {1})\Bigr{]} \tag{4}\]  
\[\ln|X(f)|^{2}=\ln|V(f)|^{2}+\ln\Bigl{[}1+\alpha^{2}+2\alpha cos2\pi f(t_{2}-t_ {1})\Bigr{]} \tag{5}\]  
Equations (6) and (7) show the spectral calculation results when wheel-flat signals are three or four, respectively. Assuming that the train running speed is constant, the time delay interval of the flat signals is equal to \(t_{2}-t_{1}=t_{3}-t_{2}=t_{4}-t_{3}=T\). As the number of wheel-flat signals increases, the multiple components of the flat cycle T in the spectrum increase one by one:  
\[|X_{3}(f)|=|V(f)|\sqrt{3+2(2 cos\omega T+cos2\omega T)} \tag{6}\]  
\[|X_{4}(f)|=|V(f)|\sqrt{4+2(3 cos\omega T+2 cos2\omega T+COS3\omega t)} \tag{7}\]  
Figure 1 shows the spectrums by periodic functions that are obtained converting four impulse signals to frequency of the flat cycle period (T = 0.243). The spectrums are composed of multiples of the flat cycle T as shown in Equation (7). The fundamental frequency in Figure 1 is 0.2 Hz, and the magnitude of three cosine waveforms remains constant without attenuation. However, the magnitudes for waveforms are decreased and distorted with the frequency intervals shortened. Because frequency intervals have a large effect on periodic components of the spectrum like the above example, the accuracy of cepstrum data becomes worse.