# Carrier Phase Dual One-Way Ranging Method Based on a Frequency Hopping Signal
## 4 Performance Analysis of the FH-DOWR Model
### The Combinational Error of FH-DOWR  
1. Error caused by the clock drift error  
In the transmitter and receiver of the FH-DOWR system, high-precision and high-stability atomic clocks are equipped on satellites. However, there are still clock reference deviations (clock offsets) and clock frequency deviations (frequency offsets) between multiple satellites. Clock bias will result in different local pseudorange measurement times for satellite B and satellite A. This leads to ranging and time comparison errors. Taking the measurement of the relative distance between satellite B and satellite A as an example, we assume that the nominal value of the clock reference frequency between satellite B and satellite A is \(f\), the difference in frequency is \(\Delta f\), and the clock difference between satellite B and satellite A is \(\Delta t\). Equation (25) calculates the distance error caused by this deviation:  
\[\Delta\rho=\frac{\Delta f}{f}\times\Delta t\times c. \tag{25}\]  
If the clock reference frequency is 10.23 MHz and if multiple satellites adopt FH-DOWR for time synchronization and calibration clock differences, then \(\Delta t\) can be restricted to less than 1 ms. However, if the desired ranging deviation caused by this term is less than 0.1 mm, then the requirement for the relative frequency difference between satellite B and satellite A is  
\[\frac{\Delta f}{f}=\frac{\Delta\rho}{\Delta t\times c}\leq 3.33\times 10^{-10}. \tag{26}\]  
Adopting an atomic clock with an accuracy and stability better than 5.0 \(\times\) 10\({}^{-10}\) as the reference frequency for satellite A and satellite B can meet the above requirements. 2. Error caused by the relative motion of satellites  
The ranging error caused by clock bias drift is similar to that caused by relative motion between satellites during the clock deviation period \(\Delta t\), and a change in relative distance will also lead to distance measurement deviation and time comparison error. The relative radial velocity between satellite B and satellite A is assumed to be \(v(t)\). The distance measurement error caused by the relative motion between satellites during the clock deviation period is calculated as (27)  
\[\Delta L=\int_{\Delta t}v(t)dt=\overline{v}\cdot\Delta t. \tag{27}\]  
\(\overline{v}\) is the average of \(v(t)\) at \(\Delta t\). Due to the clock difference \(\Delta t\) between satellite B and satellite A, we modify (2) to (28). \[ \begin{cases}T_{A}=\frac{D_{A}}{c}+\tau_{clock}+\tau_{Bt}+\tau_{AI}\\ T_{B}=\frac{\overline{U_{B}}}{c}-\tau_{clock}+\tau_{AI}+\tau_{Bt}\end{cases}  \tag{28}\]  
where \(D_{A}-D_{B}=\Delta L=\overline{v}\Delta t\). \(D_{A}\) is the distance between satellite B and satellite A at the time when satellite B sends the ranging synchronization frame. It is contained in the local pseudorange results \(cT_{A}\) of satellite A. \(D_{B}\) is the distance between satellite A and satellite B at the time when satellite A sends the ranging synchronization frame. This is containedin the local pseudoranging results \(cT_{B}\) of satellite B. Therefore, the relative distance and clock difference calculated using satellites A and B are given by (29) and (30), respectively. \[ \begin{cases}D_{\rm A}=\frac{\xi}{2}\big{[}(T_{\rm A}+T_{\rm B})-(\tau_{\rm Bt}+ \tau_{\rm Ar}+\tau_{\rm At}+\tau_{\rm Br})\big{]}+\frac{\overline{\sigma} \Delta t}{\overline{\sigma}_{\rm At}^{2}}\\ D_{\rm B}=\frac{\xi}{2}\big{[}(T_{\rm A}+T_{\rm B})-(\tau_{\rm Bt}+\tau_{\rm Ar }+\tau_{\rm At}+\tau_{\rm Br})\big{]}-\frac{\overline{\sigma}\Delta t}{2}\\ \end{cases}  \tag{29}\]  
\[ \begin{cases}\Delta t_{\rm A}=\frac{1}{2}\big{[}(T_{\rm A}-T_{\rm B})+(\tau_{ \rm Bt}+\tau_{\rm Ar}-\tau_{\rm At}+\tau_{\rm Br})\big{]}-\frac{\overline{ \sigma}\Delta t}{2\overline{\sigma}_{\rm At}^{2}}\\ \Delta t_{\rm B}=\frac{1}{2}\big{[}(T_{\rm A}-T_{\rm B})+(\tau_{\rm Bt}+\tau_{ \rm Ar}-\tau_{\rm At}+\tau_{\rm Br})\big{]}+\frac{\overline{\sigma}\Delta t}{2 \overline{\sigma}_{\rm At}^{2}}\end{cases}