# LSOS: An FG Position Method Based on Group Phase Ranging Ambiguity Estimation of BeiDou Pseudolite
## 3 Algorithmic Principle
### Group Phase Ranging Estimation
\quad\mathbf{R}_{b}= \begin{bmatrix}x_ {b}^{1}&y_{b}^{1}&z_{b}^{1}\\ x_{b}^{2}&y_{b}^{2}&z_{b}^{2}\\...&...&...\\ x_{b}^{m}&y_{b}^{m}&z_{b}^{m}\end{bmatrix}  \tag{12}\]  
The measurement equation is constructed. In the original observation information of pseudo-satellite output by receiver, the pseudo-distance can be directly obtained from the pseudo-satellite signal, and there is no ambiguity distance information. Doppler information is related to the motion state of the receiver, there is no ranging estimation error. So, if the pseudorange expressed in \(\rho_{s}\), doppler use \(f_{d}\), said the distance of the current \begin{table}
\begin{tabular}{c c c c} \hline
**2.046 MHz** & **10.23 MHz** & **20.46 MHz** & **14.322 MHz** \\ \hline (B1I, B2I) & (B1C, B2a) & (B3I, B2I) & (B1I, B1C) \\ (B1I, B3I) & (B2I, B2a) & (B1C, B2I) & - \\ (B1I, B2a) & (B3I, B2a) & (B1C, B3I) & - \\ (B1I, B2I, B2a) & (B1C, B2a, B3I) & - & - \\ (B1I, B2I, B3I) & (B1C, B2a, B2I) & - & - \\ (B1I, B2I, B1C) & (B2I, B3I, B2a) & - & - \\ (B1I, B3I, B2a) & (B1C, B2a, B3I, B2I) & - & - \\ (B1I, B3I, B1C) & - & - & - \\ (B1I, B1C, B2a) & - & - & - \\ (B1I, B2I, B1C, B2a) & - & - & - \\ (B1I, B2I, B3I, B2a) & - & - & - \\ (B1I, B2I, B3I, B1C) & - & - & - \\ (B1I B3I, B1C, B2a) & - & - & - \\ \hline \end{tabular}
\end{table} 
Table 2: The common maximum common factor frequency characteristics. time is estimated to be \(\rho_{s}(n)=\rho_{s}(n-1)+\lambda f_{d}\Delta t\), thus it is concluded that the user receiver to the location of the estimating equations  
\[\rho_{s}(k)=\|\mathbf{R}_{s}-\mathbf{r}(k-1)\|+\Delta t-\Delta T_{s}+\varepsilon \tag{13}\]  
In the formula, \(\rho_{s}(k)= \begin{bmatrix}\rho_{s}^{1}(k)\\ \rho_{s}^{2}(k)\\ \ldots\\ \rho_{s}^{n}(k)\end{bmatrix} \), \(\mathbf{k}\) is a positive integer, \(\mathbf{k}\geq 1\). \(\Delta t\) is receiver clock error. \(\Delta T_{s}\) is pseudo-satellite clock error. if \(\mathbf{r}(n)\in\mathbf{R}_{b}\), then \(\min(\rho_{s}(k)-(\|\mathbf{R}_{s}-\mathbf{r}(k)\|+\Delta t-\Delta T_{s}+ \varepsilon))\) is recorded as the selected optimal estimated position. Therefore, the transmission time of pseudolite signals in the current position can be deduced as \(t=[t_{1},t_{2},\ldots t_{n}]\). According to the interval characteristic of each signal transmission time, the whole group phase period number \(T_{c}= \begin{bmatrix}T_{1}&T_{2}&\ldots&T_{n}\end{bmatrix} \) of each transmission time is deduced. The group phase segmentation process further solves the problem of quantifying the cyclic period of the group phase, transforming the problem into the problem of phase difference estimation within the least common multiple period. It is assumed that the inter-frequency carrier phase difference between the currently acquired multi-frequency signals is  
\[\Delta\mathbf{\Phi}=[\Delta\Phi_{1}&\Phi_{2}&\ldots&\Delta\Phi_{n}] \tag{14}\]  
According to the principle of minimum resolution of group phase above, the minimum resolution value of group phase under each path can be obtained as  
\[\Delta\mathbf{\varphi}=[\Delta\varphi_{1}&\Delta\varphi_{2}&\ldots&\Delta \varphi_{n}] \tag{15}\]  
From this, \(\mathbf{N}=[N_{1}&N_{2}&\ldots&N_{n}]\), the formula for estimating the difference in the integer of the least common multiple is obtained  
\[\min(\Delta\mathbf{\varphi}\cdot\mathbf{N}-\lfloor\Delta\mathbf{\varphi} \cdot\mathbf{N}\rfloor-\Delta\mathbf{\Phi}) \tag{16}\]  
Thus, the distance value of each signal is  
\[\widehat{\mathbf{\rho}}=c\mathbf{T}_{c}+\lambda(\mathbf{N}+\Delta\mathbf{ \psi}) \tag{17}\]  
In the formula, \(\lambda\) is the signal wavelength of the lower frequency in the frequency difference. \(\Delta\mathbf{\psi}=[\Delta\varphi_{1}&\Delta\varphi_{2}&\ldots&\Delta\varphi _{n}]\), \(\Delta\psi_{n}\) is the signal phase within a cycle with low frequency in the frequency difference at the current time.