# High-Precision Pseudo-Noise Ranging Based on BOC Signal: Zero-Bias Mitigation Methods
## 5 Performance Simulation and Analysis
### Doppler Frequency Shift  
The received signal will be affected by the Doppler effect if radial relative motion between the receiving terminal and the generating terminal exists. Thus, the Doppler effect is an indispensable factor that the satellite navigation receivers should consider as the satellites in satellite navigation systems always move around the Earth. The omitting of the Doppler frequency offset of the received signal is the source that the resolution error of the digital phase discriminator is significantly smaller than the zero-bias under noncommensurate sampling conditions. The two methods of zero-bias mitigation proposed in Section 4 ignore the influence of the phase resolution error of the signal under  
Figure 8: Resolution comparison of digital discriminators for BOCs (1, 1) signals under different bandwidth conditions (\(F_{s}=11.251\) MHz, \(F_{c}=1.023\) MHz, \(F_{sc}=2.046\) MHz, \(2d_{0}^{c}=1\) chip, \(2d_{0}^{sc}=0.25\) chip, \(T_{coh}=1\) ms).  
the condition of noncommensurate sampling. The influence of Doppler shift on phase resolution is given the priority, rather than the effect of multi-frequency shift on zero-bias.  
According to Formula (6), when the number of chips in the coherent integration time is an integer, the resolution of the digital phase discriminator is \(P(k_{s},k_{c})\). According to the analysis in [7], the approaching of Doppler frequency to the magnitude of the reference frequency \(F_{0}\) could exert an impact on the phase resolution. Figure 10 shows the simulation results of the effect of Doppler frequency on the phase resolution of the digital code discriminator. Typically, a low-speed satellite navigation receiver has a code Doppler frequency less than 1 Hz. The Doppler frequency is set in a range of \(\pm\)1 KHz in the simulation. When the coherent integration time of the digital code phase discriminator is set to 1 ms, the reference frequency \(F_{0}\) is 1 KHz. In this case, only when the Doppler frequency reaches the order of 100 Hz, the resolution of the discriminator will be exerted a large impact. When the Doppler frequency is close to 100 Hz, the resolution of the digital discriminator changes significantly.  
For the general case, assuming that the Doppler frequency is less than 1 Hz, so the effect of the digital code discriminator phase resolution can be ignored. With this assumption, the standard deviation of the zero-bias of code phase discriminator can be corrected to:  
Figure 10: The phase resolution of the digital code discriminator of the BOCs (1, 1) signal varies with the Doppler frequency (\(F_{s}=10\) MHz, \(F_{c}=1.023\) MHz, \(F_{sc}=2.046\) MHz, \(2\delta_{0}^{c}=1\) chip, \(2\delta_{0}^{sc}=0.25\) chip, \(T_{coh}=1\) ms).  
Figure 9: Zero-bias of digital code phase discriminator for BOCs (1, 1) signals at different bandwidths (\(F_{c}=1.023\) MHz, \(F_{sc}=2.046\) MHz, \(2\delta_{0}^{c}=1\) chip, \(2\delta_{0}^{sc}=0.25\) chip, \(T_{coh}=1\) ms).  
\[\sigma_{S_{d}^{c}}=\frac{\sqrt{(F_{c}+F_{d})T_{coh}\cdot\left|\overline{c}_{s}^{d}-2 \cdot round\left(\overline{c}_{s}^{d}/2\right)\right|}}{\left(1+2d_{0}^{c} \right)\overline{F}_{s}T_{coh}}, \tag{20}\]  
where \(\overline{c}_{s}^{d}\)\(\left(=2d_{0}F_{s}/(F_{c}+F_{d})\right)\) represents the average number of sampling points of the composite signal waveform that considers the Doppler code effect.  
Figure 11 depicts the code Doppler effect influence on the zero-bias of the digital code phase discriminator of the BOCs (1, 1) signal. The simulation selects three noncommensurate sampling frequencies 10.231, 10.741, and 11.251 MHz, which corresponds to weak, medium and severe zero-bias respectively. The simulation results show that the simulation results of zero-bias are consistent with the theoretical values under different Doppler code frequency offset conditions. When the Doppler frequency is smaller than 1 Hz, the zero-bias variation with the Doppler frequency is not large.