# Fast Optimization of the Installation Position of 5G-R Antenna on the Train Roof
## 3 A Fast Prediction Model for the Coupling Coefficient  
### Setting of Prediction Model for the Coupling Coefficient  
Table 3 shows the maximum coupling coefficients of three kinds of roof antennae and their corresponding installation positions, which are drawn as a schematic diagram, as shown in Figure 2d. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Installation Position** & **Port of Monopole Antenna** & **Port of Monopole Antenna** \\ **(X, Y) (m)** & **(1.268 GHz)** & **(2 GHz)** \\ \hline (3, 0) & \(-43.52\) dB & \(-50.21\) dB \\ (5, 0) & \(-41.82\) dB & \(-58.55\) dB \\ (3, 0.5) & \(-49.27\) dB & \(-63.46\) dB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the two coupling coefficients. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Position (X, Y) (m)** & **Frequency (GHz)** & **Coupling Coefficient (dB)** \\ \hline 1, 0 & 0.9 & \(-59.9\) \\ 25, 0 & 0.9 & \(-65.4\) \\ 24, 1 & 2.0 & \(-63.8\) \\ 25, 1 & 2.0 & \(-53.7\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Partial data of coupling coefficient. As can be seen from Table 3, when the operating frequency of the roof antenna is 2 GHz and the coupling coefficient reaches the maximum and minimum values, the installation co-ordinates of the roof antenna are (3, 0.4) and (15, 0.2), which are similar to the attenuation characteristics of the radiation field of the pantograph arcing; that is, the attenuation of the electric field intensity is small at the place far from the pantograph arcing. This is because the electric field strength here is mainly the result of the superposition of the radiation of the contact line and the reflection of the roof [2]. The performance of the antenna is greatly affected by the surrounding environment. Due to the influence of the roof metal plane reflection, the antenna pattern of the roof antenna will change greatly compared with that in free space. In addition, with different installation positions of the roof antenna, the antenna pattern and performance parameters (such as the main lobe magnitude and the main lobe direction) will also change [16]. Figure 3 shows the antenna pattern of the roof antenna at two installation positions (coupling coefficient(Max) and coupling coefficient(Min)) before and after optimization. Due to space limitations, only the antenna pattern of the roof antenna (f = 2 GHz) is displayed. Table 4 shows two antenna performance parameters (the main lobe magnitude (MLM) and main lobe direction (MLD) at six installation positions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Frequency**} & \multicolumn{2}{c}{**Coupling Coefficient (Max)**} & \multicolumn{2}{c}{**Coupling Coefficient (Min)**} \\ \cline{2-7}  & **Position** & **MLA (dBi)** & **MLD (\({}^{\circ}\))** & **Position** & **MLA (dBi)** & **MLD (\({}^{\circ}\))** \\ \hline
0.9 & 20, 0.4 & 4.96 & 71.0 & 14, 0.8 & 5.24 & 62.0 \\
1.5 & 25, 0.1 & 4.98 & 61.0 & 15, 0.8 & 5.00 & 65.0 \\
2.0 & 3, 0.4 & 6.81 & 65.0 & 15, 0.2 & 6.73 & 50.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the performances of roof antennae at different installation positions. Figure 3: Antenna pattern of the roof antenna with two installation positions (f = 2 GHz) (red line is the pattern of the radiation field strength of the antenna changing with direction, and the interval between the two blue lines is the main lobe width): (**a**) coupling coefficient (Max); and (**b**) coupling coefficient (Min). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Frequency**} & \multicolumn{2}{c}{**Coupling Coefficient (Max)**} & \multicolumn{2}{c}{**Coupling Coefficient (Min)**} & \multirow{2}{*}{**Difference**} \\ \cline{2-2} \cline{4-5}
**(GHz)** & **Position (X, Y)** & **Value (dB)** & **Position (X, Y)** & **Value (dB)** & **Value (dB)** \\ \hline
0.9 & 20, 0.4 & \(-\)50.10 & 14, 0.8 & \(-\)69.77 & 19.67 \\
1.5 & 25, 0.1 & \(-\)52.75 & 15, 0.8 & \(-\)73.64 & 20.89 \\
2.0 & 3, 0.4 & \(-\)30.4 & 15, 0.2 & \(-\)56.36 & 25.96 \\ \hline \hline \end{tabular}
\end{table}