# Train Wheel Condition Monitoring via Cepstral Analysis of Axle Box Accelerations
## 2 Cepstral Analysis and Navewumber Domain  
The power cepstrum \(C_{p}(\widetilde{x})\) can be defined as the power spectrum of the logarithm of the power spectrum of a function \(f(x)\):  
\[C_{f}(\widetilde{x})=|\mathcal{F}^{-1}\{\log(|\mathcal{F}\{f(x)\}|^{2})\}|^{2}. \tag{1}\]  
Here, the Fourier transform \(\mathcal{F}\{\cdot\}\) is calculated by using the fast Fourier transform (FFT) algorithm. The application of the forward FFT or the inverse FFT to the logarithm of the power spectrum in Equation (1) provides the same result apart from a scaling factor. The independent variable \(\widetilde{x}\) has the same dimension as \(x\) and is originally called quefrency to indicate that it is the inverse of the frequency, which is the case for the cepstrum of a time series. Accordingly, the independent variable of the cepstrum of data in the spatial domain is dimensionally a distance. Following the nomenclature introduced by [PERSON] et al. [21], we call the independent variable of the cepstrum navewumber when its dimension is a distance, since in this case it is the inverse of the wavenumber. Furthermore, we call the cepstral domain navewumber domain. In the following the term cepstrum is always used for the power cepstrum.  
The cepstrum can be used to convert a convolution into the addition of the individual components and thus a complicated deconvolution procedure can be performed by simply subtracting the undesired components in the cepstral domain. This procedure is called homomorphic deconvolution [26]. According to Equation (1), the cepstrum of a continuous function \(y(x)\) consisting of two components \(s(x)\) and \(r(x)\) coupled via convolution \([*]\) can be calculated by the following sequence of mathematical operations:  
1. The power spectrum \(\left|Y(k)\right|^{2}\) of the function \[y(x)=s(x)*r(x),\] (2) is calculated: \[\left|Y(k)\right|^{2}=\left|S(k)\right|^{2}\cdot\left|R(k)\right|^{2},\] (3) so that the convolution transforms to a simple multiplication in the Fourier domain.
2. The logarithm of the power spectrum is taken: \[\log(\left|Y(k)\right|^{2})=\log(\left|S(k)\right|^{2})+\log(\left|R(k)\right| ^{2}).\] (4) Thus, the components are coupled by addition.
3. Applying the inverse Fourier transform to the logarithmic power spectrum \(\log(\left|Y(k)\right|^{2})\) and finally squaring the results yield the cepstrum of \(y(x)\): \[C_{y}(\widetilde{x})=C_{s}(\widetilde{x})+C_{r}(\widetilde{x})+\text{ cross-product term}.\] (5)  
Due to the linearity of the Fourier transform, the components in the cepstrum are still coupled by addition. The final squaring operation produces a cross product term.  
However, if the cepstra of \(s\) and \(r\) occupy different navewumber ranges, this term can be omitted [27] and Equation (5) becomes  
\[C_{y}(\widetilde{x})=C_{s}(\widetilde{x})+C_{r}(\widetilde{x}). \tag{6}\]  
The cross-terms can also be avoided if the final squaring operation is not included in Equation (1).  
Periodic components of the logarithmic power spectrum are reduced to series of spikes (Dirac delta functions) in the cepstral domain. [PERSON] [27] found out that if the spectrum of a signal is smooth it maps around the origin in the navewumber domain, while the cepstrum of a periodic impulse sequence, as excited by a wheel irregularity, is also an impulse sequence with the same period. This means that in the cepstrum of ABA data, the component of a wheel defect can simply be separated from the component of the transmission path, i.e., the cepstra of the impulse responses of the track, wheel and sensor. Hence, bypassing the transmission path makes trend analyses robust towards changes of the mechanical structures of the components, which influence the signal transmission. This property makes the cepstrum a promising tool for wheel monitoring.