# BDS-3 Triple-Frequency Timing Group Delay/Differential Code Bias and Its Effect on Positioning
## 2 Data and Methods
### Triple-Frequency Satellite TGD/DCB Correction Model for BDS-3
#### 2.3.2 DCB Correction Model with Precise Satellite Clock  
Unlike the broadcast ephemeris, the UCDs included in the precise clock shown in Equation (15) imply that the DCB correction need not be considered when using the B1I/B3I dual-frequency IF combination. For the IF1213 model, any of the dual-frequency IF combination corrections can be expressed as:  
\[\overline{b}_{IF_{ij}}^{s,pre}=\frac{f_{i}^{2}\cdot b_{i}^{s}-f_{j}^{2}\cdot b _{j}^{s}}{f_{i}^{2}-f_{j}^{2}}-\frac{f_{B1I}^{s}\cdot b_{B1I}^{s}-f_{B3I}^{2} \cdot b_{B3I}^{s}}{f_{B1I}^{2}-f_{B3I}^{2}} \tag{23}\]  
then:  
\[\left\{\begin{array}{l}\overline{b}_{IF_{B1I24}}^{s,pre}=\frac{f_{B24}^{2} \cdot(DCB_{C2I-CGI}-DCB_{C1X-CGI}+DCB_{C1X-C5X})}{f_{B1I}^{2}-f_{B24}^{2}}- \frac{f_{B3I}^{2}\cdot DCB_{C2I-CGI}}{f_{B1I}^{2}-f_{B3I}^{2}}\\ \overline{b}_{IF_{B1I3I}}^{s,pre}=0\\ \overline{b}_{IF_{B1C24}}^{s,pre}=\frac{f_{B24}^{2}\cdot DCB_{C1X-C5X}}{f_{B1C }^{2}-f_{B24}^{2}}-\frac{f_{B1I}^{2}\cdot DCB_{C2I-CGI}}{f_{B1I}^{2}-f_{B3I}^ {2}}+DCB_{C1X-C6I}\\ \overline{b}_{IF_{B1C3I}}^{s,pre}=\frac{f_{B1C}^{2}\cdot DCB_{C1X-CGI}}{f_{B1C }^{2}-f_{B3I}^{2}}-\frac{f_{B1I}^{2}\cdot DCB_{C2I-CGI}}{f_{B1I}^{2}-f_{B3I}^ {2}}\\ \overline{b}_{IF_{B1I24}}^{s,pre}=\frac{f_{B24}^{2}\cdot DCB_{C2I-CGI}-DCB_{C1 X-CGI}+DCB_{C1X-C7Z})}{f_{B1I}^{2}-f_{B24}^{2}}-\frac{f_{B3I}^{2}\cdot DCB_{C2I-CGI}}{f_{B1I}^{2}-f_{B3I}^ {2}}\\ \overline{b}_{IF_{B1C24}}^{s,pre}=\frac{f_{B24}^{2}\cdot DCB_{C1X-C7Z}}{f_{B1 C}^{2}-f_{B24}^{2}}-\frac{f_{B1I}^{2}\cdot DCB_{C2I-CGI}}{f_{B1I}^{2}-f_{B3I}^ {2}}+DCB_{C1X-C6I}\end{array}\right. \tag{24}\]  
The DCB of the IF123 model after correcting for the satellite clock offset using precise clock products can be expressed as:  
\[\overline{b}_{IF_{jk}}^{s,pre}=e_{i}\cdot b_{i}^{s}+e_{j}\cdot b_{j}^{s}+e_{ k}\cdot b_{k}^{s}-\frac{f_{B1I}^{s}\cdot b_{B1I}^{s}-f_{B3I}^{2}\cdot b_{B3I}^{ s}}{f_{B1I}^{2}-f_{B3I}^{2}} \tag{25}\]  
then  
\[\left\{\begin{array}{l}\overline{b}_{IF_{B1I24}B3I}^{s,pre}=(\gamma-e_{k}) \cdot DCB_{C2I-CGI}-e_{j}\cdot(DCB_{C2I-CGI}-DCB_{C1X-C6I}+DCB_{C1X-C5X})\\ \overline{b}_{IF_{B1C24}B3I}^{s,pre}=-e_{j}\cdot DCB_{C1X-C5X}-e_{k}\cdot DCB_{ C1X-C6I}+\gamma\cdot DCB_{C2X-C6I}+(DCB_{C1X-C6I}-DCB_{C2I-CGI})\\ \overline{b}_{IF_{B1I24}B3I}^{s,pre}=(\gamma-e_{k})\cdot DCB_{C2I-CGI}-e_{j} \cdot(DCB_{C2I-CGI}-DCB_{C1X-C6I}+DCB_{C1X-C7Z})\\ \overline{b}_{IF_{B1C24}B3I}^{s,pre}=-e_{k}\cdot DCB_{C1X-C7Z}-e_{k}\cdot DCB_{ C1X-C6I}+\gamma\cdot DCB_{C2I-CGI}+(DCB_{C1X-C6I}-DCB_{C2I-CGI})\\ \end{array}\right. \tag{26}\]  
where \(\gamma=-\frac{f_{B3I}^{2}}{f_{B1I}^{2}-f_{B3I}^{2}}\).  
Additionally, the UC123 DCB correction can be expressed as:  
\[\left\{\begin{array}{l}\overline{b}_{UCB_{B1I}}^{s,pre}=-\frac{f_{B3I}^{2} \cdot DCB_{C2I-CGI}}{f_{B1I}^{2}-f_{B3I}^{2}}\\ \overline{b}_{UCB_{C1C}}^{s,pre}=-\frac{f_{B1I}^{2}\cdot DCB_{C2I-CGI}}{f_{B1I}^ {2}-f_{B3I}^{2}}+DCB_{C1X-C6I}\\ \overline{b}_{UCB_{24}}^{s,pre}=-\frac{f_{B3I}^{2}\cdot DCB_{C2I-CGI}}{f_{B1I}^ {2}-f_{B3I}^{2}}+(DCB_{C1X-C6I}-DCB_{C2I-CGI}-DCB_{C1X-C5X})\\ \overline{b}_{UCB_{24}}^{s,pre}=-\frac{f_{B1I}^{2}\cdot DCB_{C2I-CGI}}{f_{B1I}^ {2}-f_{B3I}^{2}}+(DCB_{C1X-C6I}-DCB_{C1X-C7Z})\\ \overline{b}_{UCB_{3I}}^{s,pre}=-\frac{f_{B1I}^{2}\cdot DCB_{C2I-CGI}}{f_{B1I}^ {2}-f_{B3I}^{2}}\end{array}\right. \tag{27}\]