# BDS-3 Triple-Frequency Timing Group Delay/Differential Code Bias and Its Effect on Positioning
## 2 Data and Methods
### Triple-Frequency Positioning Model of BDS-3
#### 2.2.1 IF1213  
Typically, the code and carrier phase observation models of dual-frequency IF combinations can be expressed as:  
\[\left\{\begin{array}{l}p_{r,I\bar{r}_{ij}}^{s}=\alpha_{ij}\cdot p_{r,i}^{s} +\beta_{ij}\cdot p_{r,j}^{s}\\ l_{r,I\bar{r}_{ij}}^{s}=\alpha_{ij}\cdot l_{r,i}^{s}+\beta_{ij}\cdot l_{r,j}^{s} \end{array}\right. \tag{2}\]  
with:  
\[\left\{\begin{array}{l}\alpha_{ij}=\frac{f_{1}^{2}}{f_{i}^{2}-f_{2}^{2}}\\ \beta_{ij}=-\frac{f_{2}^{2}}{f_{i}^{2}-f_{j}^{2}}\\ b_{I\bar{r}_{ij}}^{s}=\alpha_{ij}\cdot b_{i}^{s}+\beta_{ij}\cdot b_{j}^{s}\\ b_{r,I\bar{r}_{ij}}=\alpha_{ij}\cdot b_{r,i}+\beta_{ij}\cdot b_{r,j}\\ B_{I\bar{r}_{ij}}^{s}=\alpha_{ij}\cdot B_{i}^{s}+\beta_{ij}\cdot B_{j}^{s}\\ B_{r,I\bar{r}_{ij}}=\alpha_{ij}\cdot B_{r,i}+\beta_{ij}\cdot B_{r,j}\end{array}\right. \tag{3}\]  
where \(\alpha_{ij}\) and \(\beta_{ij}\) are the frequency factors of the dual-frequency IF combination; \(b_{r,I\bar{r}_{ij}}\) and \(b_{I\bar{r}_{ij}}^{s}\) are the receiver and satellite dual-frequency IF combination UCDs, respectively; and \(B_{r,I\bar{r}_{ij}}\) and \(B_{I\bar{r}_{ij}}^{s}\) are the receiver and satellite dual-frequency IF combination UPDs, respectively.  
For three different frequencies, \(i\), \(j\) and \(k\), after correction of the satellite clock offset, two mutually independent IF combinations can be formed by Equation (2), and IF1213 can be expressed as:  
\[\left\{\begin{array}{l}p_{r,IF_{ij}}^{s}=u_{r}^{s}\cdot x+dt_{r,IF_{ij}}+m_{r}^ {s}\cdot Z_{r}-\overline{b}_{IF_{ij}}^{s}+\varepsilon_{r,IF_{ij}}^{s}\\ p_{r,IF_{ik}}^{s}=u_{r}^{s}\cdot x+dt_{r,IF_{ij}}+IFB_{IF_{1213}}+m_{r}^{s} \cdot Z_{r}-\overline{b}_{IF_{ik}}^{s}+\varepsilon_{r,IF_{ik}}^{s}\\ l_{r,IF_{ij}}^{s}=u_{r}^{s}\cdot x+dt_{r,IF_{ij}}+m_{r}^{s}\cdot Z_{r}+ \overline{N}_{IF_{ij}}^{s}+\varepsilon_{r,IF_{ij}}^{s}\\ l_{r,IF_{ik}}^{s}=u_{r}^{s}\cdot x+dt_{r,IF_{ij}}+m_{r}^{s}\cdot Z_{r}+ \overline{N}_{IF_{ik}}^{s}+\varepsilon_{r,IF_{ik}}^{s}\end{array}\right. \tag{4}\]  
with:  
\[\left\{\begin{array}{l}dt_{r,IF_{ij}}=dt_{r}+b_{r,IF}=dt_{r}+(\alpha_{ij} \cdot b_{r,i}+\beta_{ij}\cdot b_{r,j})\\ IFB_{IF_{1213}}=\beta_{ij}\cdot(DCB_{r,ij}-DCB_{ij}^{s})-\beta_{ik}\cdot(DCB_{ r,jk}-DCB_{ik}^{s})\\ \overline{N}_{IF_{ij}}^{s}=\alpha_{ij}\cdot\overline{N}_{r,i}^{s}+\beta_{ij} \cdot\overline{N}_{r,j}^{s}+B_{r,IF_{ik}}-B_{IF_{ij}}^{s}+b_{IF}^{s}-b_{r,IF} \\ \overline{N}_{IF_{ik}}^{s}=\alpha_{ik}\cdot\overline{N}_{r,i}^{s}+\beta_{ik} \cdot\overline{N}_{r,k}^{s}+B_{r,IF_{ik}}-B_{IF_{ik}}^{s}+b_{IF}^{s}-b_{r,IF} \end{array}\right. \tag{5}\]  
where \(dt_{r,IF_{ij}}\) is the receiver clock offset that absorbs the dual-frequency IF receiver's UCDs; \(\overline{b}_{IF_{ij}}^{s}\) and \(\overline{b}_{IF_{2i}}^{s}\) are the satellite UCDs for the \(i/j\) and \(i/k\) IF combinations after corrected satellite clock offset, respectively; \(IFB_{IF_{1213}}\) is the IF1213 IFB parameter due to different frequency combinations of UCDs. When the satellite DCB is corrected, \(DCB_{ij}^{s}=b_{i}^{s}-b_{j}^{s}\) and \(DCB_{ik}^{s}=b_{i}^{s}-b_{k}^{s}\) will be eliminated. Receiver DCB can be expressed as \(DCB_{r,ij}=b_{r,i}-b_{r,j}\) and \(DCB_{r,ik}=b_{r,i}-b_{r,k}\). \(\overline{N}_{IF_{ij}}^{s}\) and \(\overline{N}_{IF_{ik}}^{s}\) represent \(i/j\) and \(i/k\) IF combinations floating ambiguity, respectively, and \(b_{IF}^{s}\) is the satellite UCDs included in the clock product.