## 4 Simulation Results and Analysis  
### Monte Carlo Simulations  
To achieve a comprehensive assessment of the proposed bit synchronization scheme, Monte Carlo simulations are carried out. The simulation platform is shown in Figure 7, which consists of BeiDou BII IF signal generation and software receiver baseband process. The parameters used in Monte Carlo simulations are provided in Table 2.  
The concrete steps of the BeiDou BII IF signal generation simulation are as follows. 1. Position and velocity of satellites are resolved based on stored ephemeris parameters. 2. The time delay variable is calculated according to the position and velocity of satellites and the trace. Figure 5: Data signs differential coherent algorithm with long delay time. Figure 6: The amplitude attenuations of ML and DCML bit synchronization algorithms due to frequency deviation. The blue line is for ML and the red line is for DCML. On the basis of BeiDou signal structure, carrier, ranging code and NH code are generated. 4. The BeiDou IF signal is generated according to the time delay variable and signal modulation. 5. Additional white Gaussian noise is generated according to the CN0 settings. 6. Every sampled point of the signal is quantized and stored in two bits. The signal spectrum distribution of the IF samples is shown in Figure 8.  
In the software receiver section, the proposed differential coherent bit synchronization algorithm is after conventional signal acquisition and tracking. Repeated acquisition based on 1-2 ms of coherent integration is used for signal acquisition, and DLL and PLL are used for signal tracking. To prove the reliability of the proposed algorithm, tests are done to evaluate the sensitivity of different CN0s. Using Monte Carlo simulations, additional white Gaussian noise is generated for each trial, and 1000 trials are used for each probability. Monte Carlo simulations of different algorithms are carried out for comprehensive evaluation. The first simulation tests the effect of frequency deviation on the conventional maximum likelihood bit synchronization algorithm. The detection probability is plotted as a function of the CN0s in Figure 9. Without frequency deviation, the conventional ML bit synchronization algorithm has good performance in weak signal environment. The detection probability is 0.92 at CN0 of 23 dB-Hz. However, with the increase of frequency deviation, the detection probability attenuates seriously. When the frequency deviation is 50 Hz, the ML method is nullified. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sections** & **Parameters** & **Values** \\ \hline \multirow{4}{*}{Signal Generation} & IF frequency & 4.1304 MHz \\  & sampling frequency & 16.3676 MHz \\  & visible satellites & 6, 7, 11 and 12 \\  & CN0 & 20–40 dB-Hz \\ \hline \multirow{4}{*}{Software Receiver} & DLL bandwidth & 2 Hz \\  & PLL bandwidth & 40 Hz \\ \cline{1-1}  & frequency deviation & \(0,25,50\) Hz \\ \cline{1-1}  & differential coherent delay time & 20 ms \\ \cline{1-1}  & differential coherent integration time & 40 ms \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters used in the Monte Carlo simulations. Figure 7: The diagram of the platform for Monte Carlo simulations. Therefore, \(\sum\limits_{k=1}^{20}\text{sinc}[\delta f_{d}(k)T_{c}]\text{sinc}[\delta f_{d}(k- 20m)T_{c}]\) in the Equation (19) is much smaller than \(\text{sinc}[\delta f_{d}(k)T_{coh}]\) in the Equation (9). Therefore, the proposed bit synchronization algorithm should outperform the conventional ML algorithm with a frequency deviation of an integral multiple of 50 Hz. The second simulation evaluates the effect of frequency deviation on the proposed bit synchronization algorithm. The detection probability is plotted as a function of the CN0s in Figure 10. With the increase of frequency deviation, the detection probability attenuates too, but not seriously. When the frequency deviation is 50 Hz, the detection probability is 0.72 at CN0 of 36 dB-Hz. Figure 8: Signal spectrum distribution of the BeiDou B1 IIF signals.