## 4 Analysis, Comparison and Discussion of the Results
\left\{ \begin{array}{l}\left\{\frac{\left[\sum_{j=1}^{N}\left(x_{j -}\overline{x}\right)\cdot\cos\frac{2\pi\left(j-\tau\right)}{T}\right]^{2}}{ \sum_{j=1}^{N}\cos^{2}\frac{2\pi\left(j-\tau\right)}{T}}\right.\\ \left.+\frac{\left[\sum_{j=1}^{N}\left(x_{j}-\overline{x}\right)\cdot\sin\frac{ 2\pi\left(j-\tau\right)}{T}\right]^{2}}{\sum_{j=1}^{N}\sin^{2}\frac{2\pi\left( j-\tau\right)}{T}}\right\}\end{array} \right. \tag{2}\]  
where the parameter \(\tau\) is defined in the time lag equation as [64]:  
\[\tan\left(\frac{4\pi\tau}{T}\right)=\frac{\sum_{j=1}^{N}\sin\left(\frac{4\pi \tau_{1}}{T}\right)}{\sum_{j=1}^{N}\cos\left(\frac{4\pi\tau_{1}}{T}\right)} \tag{3}\]  
where:  
\(N\) is the data block size;  
\(t_{i}\) is the measured drift time \(i\);  
\(\overline{x}\) is the mean displacement calculated using the expression \(\overline{x}=\frac{1}{N}\cdot\sum_{i=1}^{N}x_{i}\);  
\(\sigma^{2}\) is the standard deviation calculated using the expression \(\sigma^{2}=\frac{1}{N-1}\cdot\sum_{i=1}^{N}\left(x_{i}-\overline{x}\right)^{2}\). In the graphs in Figure 14, the first shows the measured signal, while the second shows the calculated waveform of the associated Lomb-Scargle periodogram and Lomb-Scargle spectral approximation in the frequency range from 0 to 1 Hz. The third graph represents the power spectrum A(f) for the total Lomb-Scargle signal, where A(f) represents P(T) and is calculated from Equation (2). The maximum associated values and absolute differences are shown graphically in Figure 15 and Table 8.  
As part of the second analysis of the signals from the surveys, we used SignalLab Sigview. Here, the analysis was entirely limited to the consideration of the complete source signals from the surveying equipment. Using commands, we performed an FFT analysis for each input signal, but this time we were interested in the overall frequency spectrum and the peak values of the entire source signal. Finally, based on the results of the FFT analysis and the plotted graphical waveforms, the corresponding dominant frequency and maximum amplitude were determined for each input signal. In summary, an analysis of the geodetic measurements was thus performed with the example of the general frequency spectrum and the example of the corresponding Lomb-Scargle spectral approximations in the selected range [65]. Based on the presented and analysed responses of the structure to certain loading events and the spectral analysis of the individual signals, the final practical value was obtained. Starting from the previously compared values of the vertical displacements, it follows that, when measuring the response of the structure, the mutual difference is relatively negligible, which indicates the possibility of relatively easy mutual alignment in practically all the events analysed. In almost all load cases, when analysing the results obtained with the RTS TS50, we observed a lower sampling frequency compared to the declared value, which indicated the capability of measurements up to 30 Hz when using GeoCOM/ZG. The average sampling frequency across the whole of the experimental work was 16 Hz. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Vertical Displacement (mm)**} & \multicolumn{4}{c}{**Lomb Scargle Periodogram**} \\  & **Min** & **Max** & **\(\Delta\)** & **Min** & **Max** & **\(\Delta\)** & \\ \hline TS50 & \(-\)5.03 & 1.21 & 6.23 & \(-\)47.7113 & \(-\)22.7398 & 24.9715 & 0.005322 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Numerical differences of the Lomb–Scargle spectral approximation in the frequency range from 0 to 1 Hz. The monitoring of the dynamic effect on the displacement of the measuring points on the railway threshold was based on the acquisition of signals for the determination of the vertical displacement. The monitoring was carried out in the field and in the laboratory. The results of the monitoring in the laboratory were compared with the results of the FFT analyses performed to determine the dominant frequency and amplitude.