# Research on the Impact of BDS-2/3 Receiver ISB on LEO Satellite POD
## 2 Gnss Isb  
The Tianping-1B receiver could receive the dual-frequency code and phase measurements. This paper adopts the ionosphere-free (IF) measurements combined from dual-frequency measurements to correct the ionosphere delay. Therefore, for the onboard GPS and BDS IF measurements, the observation is as follows:  
\[\left\{ \begin{array}{l}P_{IF}^{\rm{G}}=\rho^{\rm{G}}+c\cdot\left(dt_{R}^{\rm{G}} -dt_{s}^{\rm{G}}\right)+\varepsilon_{P_{IF}}^{\rm{G}}\\ P_{IF}^{\rm{G}}=\rho^{\rm{C}}+c\cdot\left(dt_{R}^{\rm{G}}-dt_{s}^{\rm{G}} \right)+\varepsilon_{P_{IF}}^{\rm{G}}\\ L_{IF}^{\rm{G}}=\rho^{\rm{G}}+c\cdot\left(dt_{R}^{\rm{G}}-dt_{s}^{\rm{G}} \right)+\lambda_{IF}^{\rm{G}}N_{IF}^{\rm{G}}+\varepsilon_{L_{IF}}^{\rm{G}}\\ L_{IF}^{\rm{C}}=\rho^{\rm{C}}+c\cdot\left(dt_{R}^{\rm{C}}-dt_{s}^{\rm{C}} \right)+\lambda_{IF}^{\rm{C}}N_{IF}^{\rm{C}}+\varepsilon_{L_{IF}}^{\rm{C}} \end{array} \right. \tag{1}\]  
The superscripts G and C denote, respectively, the GPS and BDS; \(P_{IF}\) and \(L_{IF}\) denote, respectively, the IF code measurements and phase measurements; \(\rho\) is the true distance from the receiver to the navigation satellite; \(dt_{R}\) denotes the receiver clock errors of GPS and BDS; \(dt_{S}\) denotes the navigation satellite clock error; \(\lambda_{IF}\) and \(N_{IF}\) denotes the dual-frequency IF wavelength and ambiguity value and \(\varepsilon_{P_{IF}}\) and \(\varepsilon_{L_{IF}}\) denote the IF code and phase noise. The different GNSS signals have different hardware delay biases of signal transmission in the receiver. In the data processing, the hardware delay bias would be assimilated by the receiver clock error. The receiver clock error estimated is expressed as follows:  
\[dt_{R}=dt_{r}+dt_{hd} \tag{2}\]  
The superscript \(\rm{dt_{r}}\) denotes the clock's true error, and \(dt_{hd}\) denotes the hardware delay bias of the signal transmission in the receiver. The superscript \(dt_{R}\) denotes the GNSS receiver clock error, which contains the clock's true error \(dt_{r}\) and hardware delay bias \(dt_{hd}\).Therefore, the GPS and BDS receiver clock errors are expressed as follows:  
\[dt_{R}^{\rm{G}}=dt_{r}+dt_{hd}^{\rm{G}},\ dt_{R}^{\rm{C}}=dt_{r}+dt_{hd}^{\rm{ C}} \tag{3}\]  
The difference of \(dt_{hd}^{\rm{G}}\) and \(dt_{hd}^{\rm{C}}\) could be considered as the receiver ISB (intersystem bias) of GPS and BDS, recorded as \(B^{\rm{CG}}\):  
\[B^{\rm{CG}}=dt_{hd}^{\rm{C}}-dt_{hd}^{\rm{G}}=dt_{R}^{\rm{G}}-dt_{R}^{\rm{G}} \tag{4}\]  
Meanwhile, it is worth noting that the magnitude of the hardware delay of the carrier phase measurements is very small, and the carrier phase hardware delay could be absorbed into the ambiguity value [37]. Therefore, the ISBs of multi-GNSS receiver code measurements only are studied in this paper. In a general way, the method to deal with ISB is to set the hardware delay bias to zero. Equation (4) could be expressed as follows:  
\[B^{\rm{CG}}=dt_{R}^{\rm{G}}-dt_{R}^{\rm{G}} \tag{5}\]  
The receiver clock error of BDS could be expressed as follows:  
\[dt_{R}^{\rm{G}}=dt_{R}^{\rm{G}}+B^{\rm{CG}} \tag{6}\]  
Therefore, Equation (1) could be expressed as follow:  
\[\left\{ \begin{array}{l}P_{IF}^{\rm{G}}=\rho^{\rm{G}}+c\cdot\left(dt_{R}^{\rm {G}}-dt_{s}^{\rm{G}}\right)+\varepsilon_{P_{IF}}^{\rm{G}}\\ P_{IF}^{\rm{C}}=\rho^{\rm{C}}+c\cdot\left(dt_{R}^{\rm{G}}-dt_{s}^{\rm{G}}+B^{ \rm{CG}}\right)+\varepsilon_{P_{IF}}^{\rm{G}}\\ L_{IF}^{\rm{G}}=\rho^{\rm{G}}+c\cdot\left(dt_{R}^{\rm{G}}-dt_{s}^{\rm{G}}\right) +\lambda_{IF}^{\rm{G}}N_{IF}^{\rm{G}}+\varepsilon_{L_{IF}}^{\rm{G}}\\ L_{IF}^{\rm{C}}=\rho^{\rm{C}}+c\cdot\left(dt_{R}^{\rm{G}}-dt_{s}^{\rm{G}}+B^{ \rm{CG}}\right)+\lambda_{IF}^{\rm{C}}N_{IF}^{\rm{C}}+\varepsilon_{L_{IF}}^{\rm{C }}\end{array} \right. \tag{7}\]  
We find, both Equations (1) and (7) are equivalent to deal with ISB.