# High-Precision Pseudo-Noise Ranging Based on BOC Signal: Zero-Bias Mitigation Methods
## 5 Performance Simulation and Analysis
### Thermal Noise  
There is a little correlation between the zero-bias of the digital phase discriminator and the thermal noise in the signal on account of the distortion error caused by the sampling. Therefore, the effects of thermal noise are not considered in the signal model in Section 2 and the zero-bias statistical error analysis in Section 3. In satellite navigation receivers, the estimation of the phase is affected by thermal noise since the signal power is typically lower than noise power. In this case, thermal noise can affect the accuracy of the pseudorange measurement. In order to compare the thermal noise error and the zero-bias in the pseudorange measurement, a joint analysis is performed below.  
For the BOC signal, the thermal noise error of the subcarrier phase discriminator \(\sigma_{T}\) is:  
\[\sigma_{T}=\sqrt{\frac{d_{0}^{sc}}{2T_{coh}\cdot C/N_{0}}\Bigg{(}1+\frac{1}{ \left(1-d_{0}^{sc}\right)T_{coh}\cdot C/N_{0}}\Bigg{)}}, \tag{21}\]  
where \(C/N_{0}\) represents the carrier-to-noise ratio of the BOC signal.  
For the short-period spreading code sequence, assuming that the initial phase of the reception is unchanged, the zero-bias of the BOC signal is a fixed value depending on the sequence of the spreading code and the initial phase of the received signal. Therefore, making it possible for the fixed value to reach a maximum. For a non-periodic spreading code sequence, the zero-bias of the BOC signal at different times approximately satisfies the standard normal distribution.  
Figure 11: The zero-bias curve of the digital code phase discriminator of the BOCs (1, 1) signal with the code Doppler frequency (\(F_{c}\) = 1.023 MHz, \(2d_{0}^{c}\) = 1 chip, \(T_{coh}\) = 1 ms).  
For BOC signals based on non-periodic codes, the standard deviation \(\sigma_{D}^{2}\) of the phase measurement is  
\[\sigma_{D}^{2}=k_{\mathrm{sc}}^{2}\sigma_{S^{\prime}}^{2}+\sigma_{T}^{2}, \tag{22}\]  
where \(k_{\mathrm{sc}}\) represents the error transfer coefficient of the code tracking loop to the subcarrier tracking loop.  
According to the loop filtering theory of [PERSON] [31; 32; 33], if the phase noise obeys a normal distribution, and the original phase discrimination value has an observation error of \(\sigma_{D^{\prime}}^{2}\), the error of the obtained phase measurement value is as follows after applying a low pass filter with the bandwidth \(B_{L}\).  
\[\sigma^{2}\cong\sigma_{D}^{2}2B_{L}T_{coh}. \tag{23}\]  
Figure 12 simulates the zero-bias and thermal noise errors in the BOC (\(1,1\)) signal reception. The simulation selects sampling frequencies of 5 and 10 MHz respectively representing severe and slight sample distortions. The simulation results show that: (1) For long-period spreading code sequences, when the sampling frequency is 5 MHz and the received signal-to-noise ratio is greater than 65 dBHz, the effect of sampling distortion will be greater than thermal noise. When the sampling frequency is 10 MHz, the corresponding signal-to-noise ratio threshold is 70 dBHz. (2) For short-period spreading code sequences, when the carrier-to-noise ratio is 20.5 dBHz (5 MHz) and 30.3 dBHz (10 MHz), the maximum value of the zero-bias of the received signal is equivalent to the thermal noise error. When the carrier-to-noise ratio is 38.5 dBHz (5 MHz) and 49.5 dBHz (10 MHz), the standard deviation of the zero-bias of the received signal is comparable to the thermal noise error.