## 3 Simulation and Analysis
### Two Satellites under Synchronous Scenarios
#### 3.1.2 Results in Downlink and Analysis  
The ranging results of the I branch in downlink for a given CNR depend on the PWDF and the random noise. Because of the randomness of the noise, the compound of the true signal and the noise can be regarded as a stochastic progress. Therefore, for each execution of the simulation, the received signal can be regarded as a sample function of the stochastic progress, causing the ranging results to fluctuate slightly around the true value. This conclusion can be corroborated by Figure 7, in which \((C/N_{0})_{downlink}\) equals 75 dB-Hz and the PWDF selected is \(\alpha=20\)-dB. The five simulation results form five different curves around the true distance 10 km, and all of the ranging precisions can be accepted. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Category** & **Parameter** & **Value** \\ \hline \multirow{6}{*}{Uplink (S-band)} & Sampling frequency & 62 MHz \\  & Intermediate frequency & 15.48 MHz \\  & PN code rate & 10.23 MHz \\ \cline{1-1}  & PN code length & 10,230 \\ \cline{1-1}  & Data flow rate in I branch & 1 kHz \\ \cline{1-1}  & Data flow rate in Q branch & 20.46 kHz \\ \cline{1-1}  & Bandwidth & 20.46 MHz \\ \hline \multirow{6}{*}{Downlink (C-band)} & Sampling frequency & 320 MHz \\ \cline{1-1}  & Intermediate frequency & 46.52 MHz \\ \cline{1-1}  & PN code rate & 20.46 MHz \\ \cline{1-1}  & PN code length & 20,460 \\ \cline{1-1}  & Data flow rate in I branch & 1 kHz \\ \cline{1-1}  & Data flow rate in Q branch & 20 MHz \\ \cline{1-1}  & Bandwidth & 40.96 MHz \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Basic simulation parameters in simultaneous ranging and communication algorithm. Figure 6: Ranging result in uplink with the ranging branch only and QPSK modulation with \((C/N_{0})_{upllink}=45\) dB-Hz. To shed light on the degree of jitter under different PWDFs, Figure 8 shows the ranging results of the proposed algorithm when \((C/N_{0})_{downlink}=75\) dB-Hz with the PWDF \(\alpha\in\{0,5,10,15,10,25\}\) dB, and the simulation time is set as 10 s. To show the jitter caused by PWDF in detail, the results of the running time from 2 to 10 s are plotted in Figure 8, in which the initial tracking section is omitted. We note that the degree of unsteadiness the ranging results becomes more severe as the PWDF increases. However, this kind of relationship is limited, and the phenomenon becomes insignificant when the PWDF is sufficiently small. For instance, when the PWDFs are 0 and 5 dB, the jitters are relatively close, and their difference is not obviously. The ranging error is relevant to CNR and PWDF. To evaluate the ranging performance, we execute \(N=50\) Monte Carlo experiments, and calculated and recorded the root mean square error (RMSE) of the ranging error in downlink using Equation (23)  
\[RMSE=\sqrt{\frac{1}{N}\sum_{n=1}^{N}\sum_{t=1}^{T}\left(\hat{d}_{t}^{(n)}-d \right)^{2}}, \tag{23}\]  
where \(\hat{d}_{t}^{(n)}\) is the corresponding ranging result of the \(n\)th trial in the \(t\)th time step, and \(d\) is the true distance between the MS and SS. Figure 8: Ranging result in downlink with different PWDFs with \((C/N_{0})_{downlink}=75\) dB-Hz. Figure 7: Repetitive experiments about ranging results in downlink. According to Equation (22), as the PWDF increases, the equivalent CNR of the I branch in downlink decreases. A lower equivalent CNR of the I branch reduces the ranging precision in downlink. Thus, if the PWDF is negative infinity, the ranging precision reaches its maximum, which can be regarded as a benchmark to evaluate the ranging performance. Then, when the PWDF is 0 dB, the power of the I branch is equal to that of the Q branch. It is obvious that the power of the I branch should be lower, because of the spread spectrum gain resulting from the strong autocorrelation of the PN code. Therefore, the PWDF must be higher than 0 dB in downlink, which ensures the BER of the Q branch is in the range of acceptance.