# Carrier Phase Dual One-Way Ranging Method Based on a Frequency Hopping Signal
## 2 FH Dual One-Way Ranging in ISL
### FH-DOWR Signal Mode  
The accuracy of carrier phase ranging is greater than that of the PN code by many orders of magnitude, but the ambiguity of the carrier phase is difficult to solve. Currently, carrier phase ranging is achieved using multiple frequency carrier phase measurements. In a GNSS carrier phase measurement, since one satellite transmits and receives dual-frequency signals simultaneously, the carrier phase integer ambiguity can be resolved using the dual frequency. The signal model is shown in (3). \[s_{i}(t)=C_{i}(t)*D_{i}(t)*\cos(2\pi f_{i}t)\quad i=1,2 \tag{3}\]  
where \(s_{i}(t)\) is the transmit signal. \(C_{i}(t)\) is the PN code sequence. \(D_{i}(t)\) is the transmitted data. Based on the PN code offset, we calculate the transmit delay \(\tau\) and the pseudoranging  
Figure 1: The structure of the DOWR model. result \(\rho_{\text{PRN}}\). Based on the carrier phase measurement, we can obtain the carrier phase decimal cycle measurement results \(\Delta\varphi_{i}(i=1,2)\) of \(f_{1}\) and \(f_{2}\). The relationships between \(\rho_{\text{PRN}}\) and \(\Delta\varphi_{i}(i=1,2)\) are given in (4). \[\rho_{PRN}+c\Delta t+\varepsilon_{PRN}=\frac{\Delta\varphi_{1}}{2\pi}\frac{c}{ f_{1}}+\frac{c}{f_{1}}N_{1}+c\Delta t+\varepsilon_{f_{1}} \tag{4}\]  
\[\rho_{PRN}+c\Delta t+\varepsilon_{PRN}=\frac{\Delta\varphi_{2}}{2\pi}\frac{c}{ f_{2}}+\frac{c}{f_{2}}N_{2}+c\Delta t+\varepsilon_{f_{2}} \tag{5}\]  
In the ISL receiver, the code tracking loop can output \(\rho_{\text{PRN}}\) and the carrier tracking loop can output \(\Delta\varphi_{i}(i=1,2)\). However, \(N_{i}(i=1,2)\) are the carrier phase integer cycle measurement results, which are unknown. \(c\) is the light speed. \(\Delta t\) is the clock difference between the transmitting satellite and receiving satellite. \(\varepsilon_{PRN}\) is the pseudoranging error. \(\varepsilon_{f_{i}}(i=1,2)\) is the carrier phase measurement error. Converting (4) and (5) to (6) and (7), the following are obtained:  
\[N_{1}+N_{2}=\rho_{PRN}\bigg{(}\frac{f_{1}}{c}+\frac{f_{2}}{c}\bigg{)}+\sqrt{ \bigg{(}\frac{f_{1}}{c}\varepsilon_{PRN}\bigg{)}^{2}+\bigg{(}\frac{f_{2}}{c} \varepsilon_{PRN}\bigg{)}^{2}+\varepsilon_{f_{1}}^{2}+\varepsilon_{f_{2}}^{2} }-\frac{\Delta\varphi_{1}}{2\pi}-\frac{\Delta\varphi_{2}}{2\pi}, \tag{6}\]  
\[N_{1}-N_{2}=\rho_{PRN}\bigg{(}\frac{f_{1}}{c}-\frac{f_{2}}{c}\bigg{)}+\sqrt{ \bigg{(}\frac{f_{1}}{c}\varepsilon_{PRN}\bigg{)}^{2}+\bigg{(}\frac{f_{2}}{c} \varepsilon_{PRN}\bigg{)}^{2}+\varepsilon_{f_{1}}^{2}+\varepsilon_{f_{2}}^{2} }-\bigg{(}\frac{\Delta\varphi_{1}}{2\pi}-\frac{\Delta\varphi_{2}}{2\pi}\bigg{)}. \tag{7}\]  
Thus, the integer ambiguity of \(f_{1}\) and \(f_{2}\) is \(\frac{(N_{1}+N_{2})+(N_{1}-N_{2})}{2}\) and \(\frac{(N_{1}+N_{2})-(N_{1}-N_{2})}{2}\), respectively. However, the probability of the correct integer ambiguity is determined by the second item on the right side, which encompasses the measurement error associated with the PN code ranging and carrier phase measurement [25; 26]. This error can manifest in decimeters or even meters without processing. If the carrier frequency exceeds a certain threshold, then the method cannot be used to obtain the correct integer ambiguity resolution result. In this paper, we design the carrier phase ranging in the ISL, as shown in Figure 2.  
Figure 2: Dual-frequency DOWR with time division: (**a**) transmitting and receiving time slot and (**b**) receiving channel. The transmit signal can be expressed as (8):  
\[s(t)= \begin{cases}C(t)*D(t)*\cos(2\pi f_{1}t)&(n-1)T_{suit}<t<nT_{suit}\\ C(t)*D(t)*\cos(2\pi f_{2}t)&nT_{suit}<t<(n+1)T_{suit}\end{cases}  \tag{8}\]  
where \(s(t)\) is the transmitting signal. \(C(t)\) is the PN code sequence. Its code length is 1023, and its code frequency is 5.115 MHz. \(T_{suit}\) is the signal switching time. This method employs the same transmission and reception channels for both sending and receiving signals, thereby minimizing the utilization of hardware resources.