# LSOS: An FG Position Method Based on Group Phase Ranging Ambiguity Estimation of BeiDou Pseudolite
## 3 Algorithmic Principle
### Group Phase Ranging Estimation  
The process of group phase quantization is actually the refinement and segmentation process of signal spatial transmission. Through group phase quantization, the phase information of signal is accurately classified. The following ranging process is based on this expansion. In order to better evaluate the application of the group phase period quantization principle in the pseudo-satellite system, the group phase period of the BeiDou pseudo-satellite under multi-frequency is analyzed in combination with the common frequency points of BeiDou. A total of six commonly used frequency points are selected in the following table, as shown in Table 1.  
Combined with the maximum common factor frequency characteristics between frequencies, the first step of this paper is to analyze the maximum common factor frequency characteristics between frequencies of dual-frequency signals. Influenced by the reference frequency, the common maximum common factor frequency characteristics are shown in the combination estimation process of multiple frequencies. As shown in Table 2, four maximum common factor frequency values are obtained in total from the selected frequencies. It can be seen that the group phase quantization period after multi-frequency combination divides the signal transmission process into four segments and converts the group phase quantization period into a constant representation of distance  
\[\mathbf{S}=\left[146.5261,29.3052,20.9323\;14.6526\right] \tag{9}\]  
Normally, the standard height of the indoor space is 2.8-3.3 m, the underground parking lot is 2.5 m, the stadium, station and exhibition center is 30 m, the tunnel is 5.8 m, and the underground pipe gallery is 4.6-6 m. BeiDou pseudolite has abundant group phase quantization period segmentation characteristics, which fully meets the requirements of indoor underground space ranging. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Serial Number** & **Frequency** & **Value (MHz)** \\ \hline
1 & B1I & 1561.098 \\
2 & B2I, B2b & 1207.14 \\
3 & B3I & 1268.52 \\
4 & B2a & 1176.45 \\
5 & B1C & 1575.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of existing frequency points of the BeiDou signal. The range estimation process in detail is given below. In order to facilitate analysis, the putative \(f_{1}/f_{2}/f_{3}\) three frequency signals, and \(f_{1}<f_{2}<f_{3}\). From this, we can get the quantization period set of group phase under different frequency combinations. Then the group phase quantization period set is  
\[T= \begin{bmatrix}T_{1}&T_{2}&T_{3}&T_{4}\end{bmatrix}  \tag{10}\]  
Due to the influence of channel hardware and software delay, the pseudo-satellite signal of each channel is usually not 0. However, due to the strict time-frequency characteristics of pseudolite, it can be assumed that each launch channel has the same phase delay. In the phase comparison process, Formula (17) is the group phase quantization cycle set. \(\Delta\phi_{0}\) is the minimum period element for realizing the same phase delay again in the frequency point combination. Assume that the initial phase difference of \(\Delta\phi\), ranging error is introduced in the initial phase difference  
\[\Delta\rho=\lambda\frac{\Delta\phi}{\Delta\phi_{0}} \tag{11}\]  
Since the distance error introduced by phase difference is a constant value, it will not affect the variation of group phase quantization period in practical application. Therefore, the following distance estimation process is no longer considered alone and the default initial phase is 0. First of all, the distance estimation process and the pseudo-satellite positioning process require at least four pseudo-satellite signals. It is assumed that there are n pseudo-satellites, the position coordinate array \(\mathbf{R}_{s}\) and that the reference point coordinates of the spatial virtual grid based on the covering domain are  
\[\mathbf{R}_{s}= \begin{bmatrix}x_{s}^{1}&y_{s}^{1}&z_{s}^{1}\\ x_{s}^{2}&y_{s}^{2}&z_{s}^{2}\\...&...&...\\ x_{s}^{m}&y_{s}^{m}&z_{s}^{m}\end{bmatrix}