# Study of Rail Squat Characteristics through Analysis of Train Axle Box Acceleration Frequency
## 3 Wavelet Analysis  
### Frequency Analysis Using Wavelet Analysis  
The axis acceleration measurement in the time domain is not sufficient to detect tiny defects. Various analytical methods may be used to analyze the frequency characteristics of a signal, but not all methods provide good information. The major drawback of the short-time Fourier transform (STFT) often applied to frequency characteristics analysis is that the window size must be selected. When axis acceleration is analyzed using STFT, the time and frequency resolutions are inversely proportional if the window size is changed. For example, when the window is short, the frequency resolution decreases while the time resolution increases. The detection of different squats and rail element evaluation over a wider range require different window sizes for each defect type. Therefore, STFT is not considered a suitable method.  
Figure 2: EM-140K accelerometer.  
Therefore, we applied wavelet transform by analyzing the acceleration data in the present study and did not apply STFT analysis techniques to find the squats. This technique was used to assess the condition of the track; many previous papers have applied the wavelet transform. Representative examples include Professor [PERSON]'s team at Delft University of Technology in the Netherlands and [PERSON] in Japan. The present study was conducted using wavelet analysis, which has the advantage that the time-frequency expression is not dependent on the window size. Continuous wavelet transform (CWT) is a time-frequency analysis method in which an observed function is multiplied by a shifted and scaled wavelet function group. There are many cases and papers where wavelet transform has been applied as a technique to evaluate the state of an orbit.  
CWT is defined as in the equation below [26]:  
\[W_{n}(S)=\sum\
olimits_{n^{\prime}=0}^{N-1}x_{n}\psi^{*}\bigg{(}\frac{(n^{ \prime}-n)\delta_{t}}{s}\bigg{)} \tag{1}\]  
In Equation (1), \(x_{n}\) is a time series in which the time step is \(\delta_{t}\), and \(n\) is a time index wherein \(n^{\prime}=0,\ldots,N-1\) is a time shift operator. \(\psi\) is a mother wavelet with locally limited functions. \(\psi^{*}\) (\((n^{\prime}-n)\)\(\delta_{t}/s\)) is a wavelet derived from a mother wavelet by translation and scaling. The symbol \({}^{*}\) denotes a conjugate complex number, and \(s\) is a wavelet scale (\(s>0\)). \(W_{n}(s)\) is the wavelet coefficient.  
The wavelet scale is related to the Fourier period (or inverse frequency). According to [PERSON]'s energy conservation theorem, the energy of a wavelet transform is equal to the energy of the original signal in the time domain. The physical meaning of the CWT may be explained by the correlation in a delayed n between the original signal and the scaled wavelet. Wavelet transformation may be considered as a linear filter involving a parallel filter set [27]. The key advantage of CWT over STFT is the high time and frequency resolutions [18]. Therefore, CWT is suitable for investigation of local changes of frequency components, including the detection of structural damage and defects and identification of cracks and other abnormal phenomena [27; 28; 29]. In the present study, the Morlet function was employed as the mother wavelet.  
The Morlet function is defined as in Equation (2):  
\[\psi_{0}(\eta)=\pi^{-1/4}e^{i\eta_{0}\eta}e^{-\eta^{2}/2} \tag{2}\]  
where \(\omega_{0}\) is a dimensionless frequency. The power spectrum of a wavelet transform is the square of the wavelet coefficient, which is expressed as Equation (3):  
\[\left|W_{n}^{2}(s)\right| \tag{3}\]  
The wavelet power spectrum (WPS) is expressed in a scalogram. The vertical slice on the right side of the wavelet plot is the scale of a general spectrum. Figure 3 shows the scalogram of the acceleration signals with respect to the squats. The red color represents a high signal energy level due to impact at a squat. In the present study, the scalogram was used to define the time-frequency relationship of the squats and the axis acceleration signals.