# Analysis of Dynamic Parameters of a Railway Bridge
## 4 Covariance Model of Vibration Signal Parameters  
In the theoretical model [24; 25; 26; 27], errors of measurement of digital vibration signals are assumed to be random and systematic. The trend of measurement data of vectors is eliminated in each vector of the arrays of measurement of vibrations parameters. Time interval of dissemination of vibrations is used as one of the parameters. A random function compiled on the basis of arrays of measurement data of vibration parameters \(\varphi\) is considered stationary (in the broad sense), i.e., with the average of \(M[\varphi(t)]\rightarrow\textit{{const}}\), while the covariance function depends solely on the difference between arguments \(\tau-K_{\varphi}(\tau)\) Discrete Fourier transformation [16; 17; 18; 19; 28; 29; 30; 31; 32] may be used to process digital signals. Auto-covariance function of one data array or inter-covariance function of two arrays \(K_{\varphi}(\tau)\) is expressed as follows [78]:  
\[K_{\varphi}(\tau)=M\big{\{}\epsilon\overline{\varphi_{1}}(u)\text{-}\epsilon \overline{\varphi}_{2}(u+\tau)\big{\}}. \tag{15}\]  
or  
\[K_{\varphi}(\tau)=\frac{1}{T-\tau}\int_{0}^{T-}\Bigl{\{}\epsilon\overline{ \varphi_{1}}(u)\text{-}\epsilon\overline{\varphi}_{2}(u+\tau)\Bigr{\}}du_{ \text{,}}. \tag{16}\]  
where \(\epsilon\varphi_{1}=\varphi_{1}-\overline{\varphi_{1}}\), \(\epsilon\varphi_{2}=\varphi_{2}-\overline{\varphi_{2}}\) are the centred measurement vectors of vibration parameters \(\varphi\), after the elimination of the trend, \(u\) is the vibrations parameter, \(\tau=k\cdot\Delta\) is the variable quantum interval, \(k\) is the number of units of measure, \(\Delta\) is the value of units of measure, \(T\) is time, and \(M\) is the symbol of an average. According to the available vibration parameter measurement data, the covariance function estimate \(K^{\prime}_{\varphi}(k)\) is calculated as follows:  
\[K^{\prime}_{\varphi}(\tau)=K^{\prime}_{\varphi}(k)=\frac{1}{n-k}\sum_{i=1}^{ n-k}\epsilon\varphi_{1}(u_{i})\epsilon\varphi_{2}(u_{i+k}), \tag{17}\]  
where \(n\) is the total number of discrete intervals. Formula (14) may be applied in the form of an auto-covariance or inter-covariance function. In case of an auto-covariance function, arrays \(\epsilon\varphi_{1}(u)\) and \(\epsilon\varphi_{2}(u+\tau)\) are parts of individual arrays, and when a function is a covariance function, there are two different arrays. \begin{table}
\begin{tabular}{c c} \hline \hline
**Vector Name** & **Standard Deviation Estimate of an Individual Result \(\omega_{\varphi}\), m/s\({}^{2}\)** \\ \hline
1 & 0.001 \\
2 & 0.005 \\
3 & 0.010 \\
4 & 0.027 \\
5 & 0.379 \\
6 & 0.349 \\
7 & 0.532 \\
8 & 0.452 \\ \hline \hline  & **Parameter Name** & **Value m/s\({}^{2}\)** \\ \hline  & Estimate of standard deviation of the result the weight of which is equal to one \(\sigma^{\prime}_{\varphi}\) & 3.16\(\cdot\)10\({}^{-4}\) \\  & Estimate of standard deviation of the result the weight of which is equal to one \(\sigma^{\prime}_{\varphi}\) & 3.16\(\cdot\)10\({}^{-4}\) \\  & Estimate of standard deviation of weighted average \(\sigma^{\prime}_{\overline{\varphi}}\) & 8.5\(\cdot\)10\({}^{-9}\) \\  & Estimate of standard deviation of weighted average \(\sigma^{\prime}_{\overline{\varphi}}\) & 7.05\(\cdot\)10\({}^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Precision indicators of acceleration vectors. The estimate of the normalized covariance function is:  
\[R^{\prime}_{\varphi}(k)=\frac{K^{\prime}_{\varphi}(k)}{K^{\prime}_{\varphi}(0)}= \frac{K^{\prime}_{\varphi}(k)}{\sigma^{\prime 2}_{\varphi}}, \tag{18}\]  
where \(\sigma^{\prime}_{\varphi}\) is the estimate of standard deviation of a random function.