## 4 Analysis, Comparison and Discussion of the Results
These results also showed the consistency of the measured values. Field measurements have shown that train mass and speed have a strong influence on the intensity of vertical deformation [66]. The vertical deformation caused by a freight train can be up to three times greater than the vertical deformation caused by a passenger train. This was confirmed by the measurement of vertical accelerations, which were four times higher when a freight train passed over the same measuring point. In the laboratory measurements, we could not fully follow the tides generated by a moving train in the field, as we could not simulate the rolling effect of the train wheels, which also cause loads in the longitudinal and transverse directions. Based on the field measurements and the possibilities allowed by the measuring equipment, we set the excitation frequency to 1.5 Hz. We found that, for pulsed loads of up to 750,000 cycles, the resultant vibration velocity vector increased linearly, after which it did not increase further and stabilised. The results of the FFT frequency analysis showed that geodetic vibration monitoring is a useful and sufficiently accurate method despite the lower signal acquisition rate. The values listed are shown in Table 9.  
Figure 15: Maximum corresponding values and absolute differences of the Lomb-Scargle spectral approximation in the frequency range from 0 to 1 Hz. Measurements in most scientific and engineering fields are characterised by the periodic signal of the measured object. The signal represents an arbitrary measurable physical quantity containing information about a time-dependent physical process. The determination of the periodicity of a signal from a mathematical point of view can be achieved in a number of different ways, among which so-called spectral analysis is of great importance for processing. The most widely used and widely represented technique in the field of spectral analysis is the Fourier-transform (FT) method of computation. FT is a mathematical concept that can be used to decompose a signal into its constituent frequencies. The FT not only gives the frequencies present in the signal but also the magnitude of each frequency present in the signal. Mathematically, the definition of the basic Fourier transform of an integrable function can be represented by an integral relation [64]:  
\[X(f)=\int_{-\infty}^{\infty}x(t)\cdot e^{-j2\pi ft}dt \tag{4}\]  
where:  
\(f\) is an arbitrary real number (frequency);  
\(x(t)\) is a function of the time-dependent signal;  
\(e^{-j2\pi ft}\) is a complex harmonic signal;  
\(X(f)\) is the Fourier transform (spectrum). The fast Fourier transform (FFT) is a mathematical algorithm that calculates the discrete Fourier transform (DFT) of a given sequence. When the time function under consideration is formed of or includes a finite number of discrete objects of knowledge or periodic signals of period N, we speak of a discrete Fourier transform. Starting from the determination of the fundamental and circular frequencies of the signal, the signal can be developed into a finite series of complex exponential functions, such as [64]:  
\[x[n]=\sum_{k=0}^{N-1}X[k]e^{jnk\Omega_{0}} \tag{5}\]  
Here, in general, the complex coefficients \(X[k]\). represent the spectrum of the signal, and \(e^{jnkT\Omega_{0}}\) is again an arbitrary complex harmonic signal. The coefficients of a periodic function \(X[k]\) with period N are calculated using the inverse expression [64]:  
\[X[k]=\frac{1}{N}\cdot\sum_{k=0}^{N-1}x[n]e^{-jnk\Omega_{0}} \tag{6}\]  
The DFT can be further transformed for further necessary analyses by introducing the complex constant transformation \(X[k]\).