# Carrier Phase Dual One-Way Ranging Method Based on a Frequency Hopping Signal
## 4 Performance Analysis of the FH-DOWR Model
### Acquisition and Tracking Process Performance  
The receiving process is accompanied by signal reacquisition and retracking. For FH-DOWR, the hopping patterns are the same at the transmitter and receiver. Thus, the FH characteristics of the signal are eliminated before the acquisition process. The time requirements of different processes in signal acquisition are shown in Figure 10. The total number of clock cycles required for one acquisition process is as follows  
\[T_{\text{acq}}=T_{\text{code}}+T_{\text{FFT}}+T_{\text{IFFT}}+T_{\text{mul}}+T _{\text{Max}}+\Delta t_{\text{acq}}. \tag{20}\]  
Figure 9: Simulation scenario design. (**a**) GPS orbit. (**b**) Relative distance. (**c**) The relative Doppler.  
\(T_{\text{code}}\) is the PN code period. \(T_{\text{FFT}}\) and \(T_{\text{IFFT}}\) are the time costs of the 4096-point FFT and IFFT, respectively. \(T_{\text{mul}}\) represents the time consumption of the local FFT receiving signal. \(T_{\text{Max}}\) is the time consumption required to reach the maximum IFFT. \(\Delta t_{acq}\) is the other clock cycle cost, which is approximately 100 clock cycles. In the system, the system sampling frequency is 81.84 MHz. Thus, the acquisition time of each Doppler frequency is 0.6 ms.  
A single acquisition increases the probability of false alarms and missed alarms. The system uses a Tong detector to reduce them [32]. For the ISLs, the residual Doppler frequency search range is \(-12\) kHz to 12 kHz, and the Doppler frequency search interval is 500 Hz. Based on the above parameters, the total time for achieving reacquisition of FH DOWR signals in ISL systems can be given as (21):  
\[T_{acq\_total}=\frac{12000\text{ Hz}*2}{500\text{ Hz}}*6*T_{acq}=172.8\text{ ms}. \tag{21}\]  
The tracking loop is composed of a DLL and PLL. The former tracks the PN code and outputs the pseudoranging results. The latter tracks the carrier phase and outputs the decimal cycle carrier phase measurement value.  
The PN code frequency of the ISL is 10.23 MHz. The ranging accuracy of the receiver is dependent mainly on the thermal noise error of the DLL and the dynamic stress error caused by relative motion. The former is reflected in the random jitter of the DLL, while the latter is influenced by the code Doppler variation. The code Doppler variation has a strict coupling relationship with the carrier Doppler, and errors can be suppressed through the Doppler offset provided by the carrier loop. Therefore, the dynamic stress error can be ignored, and only the ranging error caused by thermal noise is considered [33]. The PN code ranging error is (22).  
\[\sigma_{\text{ranging}}=T_{\text{chip}}\sqrt{\frac{4F_{\text{I}}d^{2}B_{n}}{C/N _{0}}\bigg{[}2(1-d)+\frac{4F_{\text{I}}d}{T\cdot C/N_{0}}\bigg{]}} \tag{22}\]  
\(T_{\text{chip}}\) is the PN code length width. \(d\) is the code interval of the DLL. \(B_{n}\) is the bandwidth of the DLL. \(T\) is the integration time. \(F_{\text{I}}\) is the DLL correlator factor, which we chose to be 0.5. \(F_{2}\) is the DLL discriminator factor, which we chose to be 1. An increase in \(B_{n}\), \(T\), and \(C/N_{0}\) can improve the ranging accuracy.  
The thermal noise error of the PLL is shown in (23) [34]:  
\[\sigma_{PLL}=[\frac{B_{PF}}{C/N_{0}}(1+\frac{1}{2T\cdot C/N_{0}})]^{0.5}. \tag{23}\]  
Figure 10: The time consumption of the acquisition process.  
The Doppler frequency error is expressed as (24):  
\[\sigma_{Doppler}=\frac{\sqrt{2}\sigma_{PLL}}{2\pi\cdot\Delta t_{\rm doppler}}. \tag{24}\]  
\(B_{\rm PF}\) is the PLL bandwidth, chosen as 10 Hz. The integration time \(T\) is 2 ms. \(\Delta t_{\rm doppler}\) is the Doppler frequency measurement interval, which is 0.1 s. When the CNR is 53 dB-Hz, \(\sigma_{\rm PLL}=7.07\times 10^{-3}\) rad and \(\sigma_{\rm doppler}=1.55\times 10^{-4}\) Hz.