## 3 Feature Extraction Methods and Indicator Development
### Multi-Feature Extraction Method in the Time-Frequency Domain  
Track vibration is typically a nonstationary signal, and many scholars conduct time-frequency analysis for feature extraction. [PERSON] et al. [52] used short-time Fourier transform to analyze local deformation of HSR subgrade. [PERSON] et al. [53] applied wavelet energy spectrum for track defect identification. [PERSON] et al. [54] adopted improved Empirical Mode Decomposition (EMD) to achieve the track irregularities detection. [PERSON] et al. [55] applied EMD to diagnose the train wheel faults. HHT was applied for analyzing the damage-induced change in time-varying frequency and energy of rail structure [56]. To support the extraction of time-frequency features, such as local peak and energy in a frequency band, the HHT method [47], an adaptive time-frequency analysis method, was adopted. A comparative analysis between the baseline track segment without interlayer debonding and the track segment with interlayer debonding was conducted, and the performance of time-frequency features is evaluated in Section 4.4. The flow chart of HHT is displayed in Figure 6. This algorithm starts with a temporally adaptive decomposition of a signal (i.e., EMD), producing a set of intrinsic mode functions (IMFs) and a remainder. On the basis of EMD, the original signal \(x(t)\) can be expressed in the following form:  
\[x(t)=\sum_{i=1}^{n}x_{i}(t)+r(t) \tag{3}\]  
where \(x_{i}(t)\) is the \(i\)th IMF and \(r(t)\) is the remainder. The Hilbert transform of the \(x_{i}(t)\) is obtained by the following equation:  
\[y_{i}(t)=\frac{P}{\pi}\int_{-\infty}^{+\infty}\frac{x_{i}(\tau)}{t-\tau}d\tau \tag{4}\]  
where \(P\) is the Cauchy principal value, \(y_{i}(t)\) is the Hilbert transform of each IMF, and \(y(t)\) is obtained by superimposing the Hilbert transforms of all IMFs. Combining the \(x(t)\) and \(y(t)\), we can obtain the analytic signal \(z(t)\). \[z(t)=x(t)+jy(t)=a(t)e^{i\varphi(t)} \tag{5}\]  
\[a(t)=\left[x^{2}(t)+y^{2}(t)\right]^{2}\!\!,\phi(t)=\arctan(y(t)/x(t)) \tag{6}\]  
where \(a(t)\) is the instantaneous amplitude of \(x(t)\) which can reflect the energy of \(x(t)\) varies with time, and \(\varphi(t)\) is the instantaneous phase of \(x(t)\). One important property of the Hilbert transform is that if the signal \(x(t)\) is a mono-component, then the time derivative of instantaneous phase \(\varphi(t)\) is the instantaneous frequency \(\omega(t)\) of the signal \(x(t)\). \[\omega(t)=\frac{d\varphi(t)}{dt} \tag{7}\]  
Figure 6: Flow chart of HHT. when the remainder is ignored, the original signal \(x(t)\) can be expressed as:  
\[x(t)=\text{Re}{\sum_{i=1}^{n}A_{i}(t)\exp(j\int w_{i}(t)dt)} \tag{8}\]  
where \(\omega(t)\) and \(A_{i}(t)\) are functions of time. By using HHT, rich information about frequency and amplitude can be obtained for further analysis. HHT is a time-frequency analysis method that can simultaneously obtain information about the time, frequency, and amplitude of a signal. In this study, three-dimensional time-frequency spectrums by HHT, which clearly show the change in frequency and amplitude, were obtained. In order to quantify the time-frequency spectrum, the front view of the three-dimensional time-frequency diagram is also displayed. The wheel-rail coupling system produces abnormal vibration in the debonding state which leads to a change in the waveform of the vibration, so it is necessary to focus on the amplitude of the vibration signal. Therefore, the local peak was one of the time-frequency domain indicators. Since the wheel-rail vibration in field has a large randomness, local energy was selected as one of the time-frequency domain characteristic indicators to eliminate uncertainty. In addition to time-frequency features, the equation of MAC was introduced to calculate the relationship between any two dominant IMFs and generate more features. The term CAC (Equation (9)) replaces MAC to avoid confusion with the MAC in modal analysis.