# Carrier Phase Dual One-Way Ranging Method Based on a Frequency Hopping Signal
## 3 The FH Dual One-Way Ranging System
### The Carrier Phase Integer Ambiguity Resolution Algorithm  
To obtain the carrier phase ranging result, it is necessary to solve the integer ambiguity of the carrier phase. In the carrier phase of the ISL, we should constrain the range of integer ambiguity through pseudorange rounding. By measuring the pseudorange of the interstellar link, a rough estimate of the overall ambiguity \(\bar{N}_{i}\) can be obtained as follows:  
Figure 6: Carrier phase output structure.  
\[\widetilde{N}_{i}=\left[\frac{\rho_{k}}{\lambda_{i}}\right] \tag{15}\]  
where \(k\) is the ISL mark, \(i\) is the frequency mark, and \([\bullet]\) is the rounded-down mark. In the ISL measurements, the ranging accuracy without processing is 1-5 m. Thus, for signals with an RF frequency of 23 GHz, the residual integer ambiguity is within \([-267,267]\). We adopt the residual integer ambiguity resolution algorithm as follows.  
Step 1: Obtain the ranging difference of the ISL. The ranging difference measured using the carrier phase should be roughly equal to the distance measured using the PN code, as shown in (16) and Figure 7.  
\[\rho_{21}^{*}-\Delta\rho\leq N_{1}\lambda_{1}+\varphi_{21f1}\lambda_{1}/2\pi \approx N_{2}\lambda_{2}+\varphi_{21f2}\lambda_{2}/2\pi\leq\rho_{21}+\Delta\rho \tag{16}\]  
where \(\rho_{21}^{*}\) is the PN code ranging result. \(\Delta\rho\) is the PN code ranging error. \(N_{1}\) is the integer ambiguity of the \(f_{1}\) frequency signal. \(\lambda_{1}\) is the wavelength of \(f_{1}\). \(\varphi_{21f1}\) is the phase difference of the \(f_{1}\) signal between the transmitter and receiver; the value is less than \(2\pi\). In addition, \(N_{2}\), \(\lambda_{2}\), and \(\varphi_{21f2}\) are the parameters of \(f_{2}\).  
Step 2: Convert (16) to (17).  
\[N_{1}\lambda_{1}-N_{2}\lambda_{2}\approx\varphi_{21f2}*\lambda_{2}/2\pi- \varphi_{21f1}\lambda_{1}/2\pi \tag{17}\]  
In (17), the unknown parameters on right side of the equation can be output by the tracking loop of Section 3.3.  
Step 3: Search for \(N_{1}\) and \(N_{2}\) in the range of \(\left[\rho_{21}^{*}-\Delta\rho,\rho_{21}^{*}+\Delta\rho\right]\). The minimum value of \(e_{f12}\) is calculated, and \(N_{1}\) is stored.  
\[e_{f12}=\left[N_{1}\lambda_{1}+\varphi_{21f1}\lambda_{1}/2\pi\right]-\left[N_ {2}\lambda_{2}+\varphi_{21f2}*\lambda_{2}/2\pi\right] \tag{18}\]  
Step 4: Calculate the distance difference as (19) \(f_{2}\)  
\[\rho_{21}=\min(N_{1})\lambda_{1}+\frac{\left(\varphi_{21f1}\lambda_{1}\right) }{2\pi} \tag{19}\]  
Figure 7: Dual-frequency integer ambiguity resolution algorithm.  
When the SNR is low enough, it is necessary to improve the phase measurement accuracy of the carrier tracking loop through fitting. Due to the time-division method used in this article to receive signals from multiple frequency points, not all the carrier phase measurements are available. In this paper, we chose third-order polynomials to fit the measurement data. In Figure 8, the direct carrier phase output of PLL is shown in Figure 8a. The carrier phase measurement has significant jitter and cannot perform full cycle ambiguity resolution.