# Research on Improving Satellite Positioning Precision Based on Multi-Frequency Navigation Signals
## 4 Results  
The navigation receiver of the CAPS system has 48 signal channels and can receive at least 12 channels of C-band navigation signals at the same time. Assuming that the number of downlink frequencies of each visible satellite is at most 12, when the downlink frequencies are different, the maximum, minimum, and average values of PDOP in the CAPS system and GNSS system are counted respectively. The results are shown in Table 2.  
To more intuitively represent the relationship between the PDOP value and the number of downlink frequencies, the changes of the PDOP value of CAPS and GNSS with the number of frequencies are plotted in Figure 7.  
As shown in Table 2 and Figure 7, the use of multi-frequency navigation signals for satellite positioning can significantly reduce the PDOP value of the satellite navigation system. When the number of downlink frequencies is k, the PDOP value is about \(1/\sqrt{\text{k}}\) of the original PDOP value. In the CAPS system, when the number of downlink frequency of each satellite is 1, that is, when multi-frequency is not used, the maximum value of the PDOP value is 6.85, the minimum value is 6.82, and the average value is 6.83; when the downlink frequencies of each satellite is 12, the maximum PDOP value drops to 1.98, the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**The Number of**}} & \multicolumn{3}{c}{**CAPS**} & \multicolumn{3}{c}{**GNSS**} \\ \cline{2-7}  & **Max** & **Min** & **Average** & **Max** & **Min** & **Average** \\ \hline
1 & 6.85 & 6.82 & 6.83 & 2.91 & 1.17 & 1.76 \\
2 & 4.84 & 4.82 & 4.83 & 2.06 & 0.83 & 1.24 \\
3 & 3.96 & 3.94 & 3.94 & 1.68 & 0.68 & 1.01 \\
4 & 3.43 & 3.41 & 3.41 & 1.45 & 0.58 & 0.88 \\
5 & 3.06 & 3.05 & 3.05 & 1.3 & 0.52 & 0.79 \\
6 & 2.8 & 2.78 & 2.79 & 1.19 & 0.48 & 0.72 \\
7 & 2.59 & 2.58 & 2.58 & 1.1 & 0.44 & 0.66 \\
8 & 2.42 & 2.41 & 2.41 & 1.03 & 0.41 & 0.62 \\
9 & 2.28 & 2.27 & 2.28 & 0.97 & 0.39 & 0.59 \\
10 & 2.17 & 2.16 & 2.16 & 0.92 & 0.37 & 0.56 \\
11 & 2.07 & 2.06 & 2.06 & 0.88 & 0.35 & 0.53 \\
12 & 1.98 & 1.97 & 1.97 & 0.84 & 0.34 & 0.51 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistics of PDOP values of multi-frequency navigation signals in CAPS system and GNSS system. Figure 7: The change of the PDOP value with the increase of the number of frequencies. Subfigures (**a**,**b**) are corresponding to the CAPS system and the GNSS system, respectively. minimum value drops to 1.97, and the average value drops to 1.97. In the GNSS system, when the number of the downlink frequency of each satellite is 1, the maximum PDOP value is 2.91, the minimum value is 1.17, and the average value is 1.76; when the number is 12, its maximum PDOP value drops to 0.84, its minimum value drops to 0.34, and its average value drops to 0.51. Based on the above satellite constellation space layout, take the observation station as the coordinate origin, and calculate the position coordinates of the receiver in the three directions of the north, east, and zenith at each moment. When the number of downlink frequencies is different, the root means square error, improvement percentage, and standard deviation of the three directions are calculated separately. The statistical results of the CAPS system are shown in Table 3, and the statistical results of the GNSS system are shown in Table 4; the variation of statistical results with the number of downlink frequencies are shown in Figure 8.  
As shown in Tables 3 and 4 and Figure 8, when the number of downlink frequencies is \(\mathbf{k}\), the positioning precision of the system is about \(1/\sqrt{\mathbf{k}}\) of the original. In the CAPS system, the elevation-assisted method is used for the positioning calculation, and the positioning error in the height is less than 0.01 m, so it is not counted here.