## 2 System Design
### Ranging Measurement
\begin{split}\tau(t_{S})&=\frac{1}{2}\cdot\left[ \frac{\rho_{SM}(t_{S})}{c}+\frac{\rho_{MS}(t_{M})}{c}-\Delta\tau\cdot\frac{c}{ c-\mathbf{e_{MS}}^{T}\cdot\mathbf{\bar{v}_{LOS}^{S}}}-\Omega_{+}\right]\cdot c ^{\prime}\\ \Delta t&=\frac{1}{2}\cdot\left[\frac{\rho_{SM}(t_{S })}{c}-\frac{\rho_{MS}(t_{M})}{c}+\tau(t_{S})\cdot c^{\prime\prime}+\Delta \tau\cdot\frac{c}{c-\mathbf{e_{MS}}^{T}\cdot\mathbf{\bar{v}_{LOS}^{S}}}- \Omega_{-}\right]\end{split}  \tag{18}\]  
where:  
\[ \begin{split}\Omega_{+}&=\tau_{TS}+\tau_{RM}+I_{S }+d_{S}+\tau_{TM}+\tau_{RS}+\varepsilon_{FM}+I_{M}+d_{M}\\ \Omega_{-}&=\tau_{TS}+\tau_{RM}+I_{S}+d_{S}-\tau_{ TM}-\tau_{RS}-\varepsilon_{FM}-I_{M}-d_{M}\\ c^{\prime}&=2\cdot\left[\frac{c}{c-\mathbf{e_{MS}}^{T} \cdot\mathbf{\bar{v}_{LOS}^{S}}}+\frac{c}{c-\mathbf{e_{SM}}^{T}\cdot\mathbf{ \bar{v}_{LOS}^{M}}}\right]^{-1}\\ c^{\prime\prime}&=\frac{c}{c-\mathbf{e_{MS}}^{T} \cdot\mathbf{\bar{v}_{LOS}^{S}}}-\frac{c}{c-\mathbf{e_{SM}}^{T}\cdot\mathbf{ \bar{v}_{LOS}^{M}}}\end{split}  \tag{19}\]  
The LOS velocity of the satellites changes slowly in a short time interval, thus we can assume:  
\[ \begin{split}\left\{\mathbf{\bar{v}_{LOS}^{M}}=v_{LOS}^{M}(t_{M} )=v_{LOS}^{M}(t_{S})\\ \mathbf{\bar{v}_{LOS}^{S}}=v_{LOS}^{S}(t_{M})=v_{LOS}^{S}(t_{S}) \end{split}  \tag{20}\]  
Therefore, the terms \(c^{\prime}\) and \(c^{\prime\prime}\) can be given by:  
Figure 6: Diagrammatic sketch of LOS velocity. \[c^{\prime} = 1-\frac{1}{2}.\frac{\mathbf{e_{{}_{\text{MS}}}}^{{}^{T}}\cdot\overline{ \mathbf{v}_{\text{LOS}}^{{}^{\text{S}}}}+\mathbf{e_{{}_{\text{SM}}}}^{{}^{T}} \cdot\overline{\mathbf{v}_{\text{LOS}}^{{}^{\text{M}}}}}{c}-\frac{\frac{\left( \mathbf{e_{{}_{\text{MS}}}}^{{}^{T}}\cdot\overline{\mathbf{v}_{\text{LOS}}^{{}^{ \text{S}}}}-\mathbf{e_{{}_{\text{SM}}}}^{{}^{T}}\cdot\overline{\mathbf{v}_{ \text{LOS}}^{{}^{\text{M}}}}\right)^{2}}{1+\frac{1}{2}.\frac{\mathbf{e_{{}_{ \text{MS}}}}^{{}^{T}}\cdot\overline{\mathbf{v}_{\text{LOS}}^{{}^{\text{S}}}}+ \mathbf{e_{{}_{\text{SM}}}}^{{}^{T}}\cdot\overline{\mathbf{v}_{\text{LOS}}^{{}^{ \text{M}}}}}{c}}}{c}\] \[= 1-\frac{1}{2}.\frac{\mathbf{e_{{}_{\text{MS}}}}^{{}^{T}}\cdot \overline{\mathbf{v}_{\text{LOS}}^{{}^{\text{S}}}}+\mathbf{e_{{}_{\text{SM}}} }^{{}^{T}}\cdot\overline{\mathbf{v}_{\text{LOS}}^{{}^{\text{M}}}}}{c}\] \[= 1-\frac{1}{2}.\frac{\mathbf{e_{{}_{\text{MS}}}}^{{}^{T}}\cdot \left(\overline{\mathbf{v}_{\text{LOS}}^{{}^{\text{S}}}}-\overline{\mathbf{v }_{\text{LOS}}^{{}^{\text{M}}}}\right)}{c}\] \[\quad\quad\quad\quad c^{\prime} =\frac{c}{c-\mathbf{e_{{}_{\text{MS}}}}^{{}^{T}}\cdot\overline{ \mathbf{v}_{\text{LOS}}^{{}^{\text{S}}}}}-\frac{c}{c-\mathbf{e_{{}_{\text{SM} }}}^{{}^{T}}\cdot\overline{\mathbf{v}_{\text{LOS}}^{{}^{\text{M}}}}}\] \[=\frac{c\left(\mathbf{e_{{}_{\text{MS}}}}^{{}^{T}}\cdot\overline{ \mathbf{v}_{\text{LOS}}^{{}^{\text{S}}}}-\mathbf{e_{{}_{\text{SM}}}}^{{}^{T}} \cdot\overline{\mathbf{v}_{\text{LOS}}^{{}^{\text{M}}}}\right)}{\left(c- \mathbf{e_{{}_{\text{MS}}}}^{{}^{T}}\cdot\overline{\mathbf{v}_{\text{LOS}}^{{} ^{\text{M}}}}\right)}\] \[\approx\frac{\mathbf{e_{{}_{\text{MS}}}}^{{}^{T}}\cdot\overline{ \mathbf{v}_{\text{LOS}}^{{}^{\text{S}}}}-\mathbf{e_{{}_{\text{SM}}}}^{{}^{T}} \cdot\overline{\mathbf{v}_{\text{LOS}}^{{}^{\text{M}}}}}{c}\] \[=\frac{\mathbf{e_{{}_{\text{MS}}}}^{{}^{T}}\cdot\left(\overline{ \mathbf{v}_{\text{LOS}}^{{}^{\text{S}}}}+\overline{\mathbf{v}_{\text{LOS}}^{{} ^{\text{M}}}}\right)}{c}\]  
Substitution of Equations (21) and (22) into Equation (18) yields:  
\[\left[