## 3 Test Data Analysis
### Frequency Domain Analysis
#### 3.2.2 Cross-Spectral Analysis  
Cross-power spectrum, abbreviation for cross-power density spectrum, is a method to describe the degree of statistical correlation between two different signals in the frequency domain [27]. There are two stationary random signals x(t) and y(t), according to stochastic process theory, the statistical correlation between them should be expressed by their cross correlation function. Fourier transform is performed on the cross-correlation function of x(t) and y(t) to obtain the power density spectrum in the frequency domain, which is called the cross-power density spectrum, also known as the cross-spectrum. The cross-spectrum and the cross-correlation function are two different representations that describe the statistical correlation of two signals from the frequency domain and the time domain, and they are mutual Fourier transforms and are suitable for deterministic signal analysis. Therefore, it is useful to measure the input and output signals of the system, and perform the cross-spectral operation between the two.  
The \(X_{1}(f)\) and \(X_{2}(f)\) self-spectra \(G_{x_{1}x_{1}}(f)\) and \(G_{x_{2}x_{2}}(f)\) can be obtained by the Welch average periodogram method; then, the amplitude squares of \(x_{1}(k)\) and \(x_{2}(k)\) are coherent functions [28] (referred to as the coherence function) and are given by:  
\[\gamma_{x_{1}x_{2}}^{2}(f)=\frac{\left|G_{x_{1}x_{2}}(f)\right|^{2}}{G_{x_{1}x _{1}}(f)G_{x_{2}x_{2}}(f)} \tag{1}\]  
The coherence function can measure the degree of correlation between two signals in the frequency domain. The frequency band with a large coherence function contains more source signal components, and the signal-to-noise ratio of the frequency band is higher. Therefore, the coherence function and cross-power spectral-difference spectrum of the collected vibration signals of the rail, track bed, and tunnel wall can be used to estimate the characteristic frequency band of the source signal, so that the vibration correlation between different track components can be analyzed (Figure 16).  
According to Figure 16a, it can be seen that the coherence coefficient of the rail-track bed acceleration spectrum of the steel-spring-floating slab track is significantly larger than that of the ordinary monolithic track. The rail-track bed accel  
Figure 16: Comparison of acceleration cross-spectrum between various track components. (**a**) Rail-track bed; (**b**) track bed–tunnel wall.  
curve of the ordinary monolithic track has one significant peak point at 458 Hz; the steel-spring-floating slab track has two significant peak points at 429 Hz and 693 Hz, respectively. Compared with the straight line segment, the coherence coefficient amplitude of the curved segment is larger, and there are more fluctuation points at high frequencies.  
According to Figure 16b, it can be seen that the coherence curves of the track bed-tunnel wall acceleration spectrum under the four working conditions are quite different. The track bed-tunnel wall acceleration spectral coherence curve of the ordinary monolithic track has 2 significant peak points (458 Hz, 683 Hz) in the straight line segment, and the amplitude of the peak point at 683 Hz in the curved segment increases. There are two additional peak points (967 Hz, 1162 Hz, 1875 Hz). The coherence coefficient of the track bed-tunnel wall acceleration spectrum of the steel-spring-floating slab track is significantly smaller than that of the ordinary monolithic track, and its characteristic frequency bands are located at 429 Hz, 684 Hz, and 898 Hz. The characteristic frequency bands of the curved segment have changed, and are located at 459 Hz, 703 Hz, and 1191 Hz, respectively.