# LEO-Augmented GNSS Based on Communication Navigation Integrated Signal
## 2 Performance of the Navigation Signal
### Doppler Accuracy  
The accuracy of DLL for methods 1 and 3 and the accuracy of the open loop estimation for method 2 are compared to evaluate the accuracy of Doppler.  
The accuracy of open loop estimation for frequency based on differential power discriminator is shown below [17; 18].  
\[\begin{array}{l}\sigma_{f}=\sqrt{\frac{\mu_{0}}{4T_{coh}^{2}(1-\cos(2\pi f_ {b}T_{coh}))}}\\ \mu_{0}=\frac{f_{b}^{2}(1-\cos(2\pi f_{b}T_{coh}))}{(\sin c(\pi f_{b}T_{coh}))- \cos(\pi f_{b}T_{coh})^{2}}\\ \mu_{1}=\frac{1-\sin c^{2}(2\pi f_{b}T_{coh})}{2(1-\cos(2\pi f_{b}T_{coh})) \sin c^{2}(\pi f_{b}T_{coh})}\end{array} \tag{8}\]  
where \(f_{b}\) is the search interval of frequency and its value should satisfy \(f_{b}<\frac{1}{2T_{coh}}\). C/N\({}_{0}\) is the CN0 of the navigation signal.  
Figure 4: The accuracy of pseudorange versus carrier noise power spectral density rate (CN0).  
The accuracy of FLL for continuous signal is shown as follows [14],  
\[\sigma_{FLL}=\frac{1}{2\pi T_{coh}}\sqrt{\frac{4 FB_{\text{L}}}{C/N_{0}}\bigg{(}1+ \frac{1}{T_{coh}\cdot C/N_{0}}\bigg{)}} \tag{9}\]  
F is a parameter, and it is set to 1 in this paper. \(B_{L}\) is the loop noise bandwidth, and it is set to 2 Hz.  
From Equations (8) and (9), it can be seen that the accuracy of Doppler is mainly determined by \(T_{coh}\) and CN0. Another important parameter for accuracy of open loop estimation is \(f_{b}\). Therefore, the effects of \(f_{b}\) are also considered in the following analysis.  
In the following, the Doppler accuracy of open loop estimation and FLL are compared. The following parameters are set to get the comparison result,  
\(\sigma_{OL}^{1}\): \(f_{b}=30,T_{coh}=1\) ms \(\sigma_{OL}^{3}\): \(f_{b}=40,T_{coh}=10\) ms \(\sigma_{GL}^{2}\): \(f_{b}=30,T_{coh}=10\) ms \(\sigma_{FLL}\): \(T_{coh}=1\) ms  
\(\sigma_{OL}^{1}\), \(\sigma_{OL}^{2}\), and \(\sigma_{OL}^{3}\) correspond to method 2, and \(\sigma_{FLL}\) corresponds to methods 1 and 3. The comparison result is shown below.  
From Figure 5, we draw the following conclusions.  
1. Comparing \(\sigma_{OL}^{1}\) with \(\sigma_{OL}^{2}\), it can be seen that with the increment of \(T_{coh}\), \(\sigma\) decreases greatly.
2. Comparing \(\sigma_{OL}^{2}\) with \(\sigma_{OL}^{3}\), it can be seen that the effect of \(f_{b}\) to \(\sigma\) is not the same during the whole CN0 range. When CN0 is lower than 37 dBHz, \(f_{b}=40\) shows better performance, and when CN0 is higher than 37 dBHz, \(f_{b}=30\) shows better performance. The optimal choice of \(f_{b}\) is another optimization objective and it is not analyzed in this paper.
3. Comparing \(\sigma_{FLL}\) with the others, it can be seen that when \(T_{coh}\) is the same, FLL is better than open loop estimation and, if \(T_{coh}\) is increased, the accuracy of open loop estimation is close to FLL.  
Based on the analysis above, it can be concluded that for method 1 and method 3, the accuracy of Doppler depends only on CN0 of the navigation signal, and for method 2, the accuracy of Doppler can be close to method 1 and method 3 when the receiving parameters are chosen optimally.  
Therefore, the difference of Doppler accuracy among the three methods will be emitted in the following.  
Figure 5: The accuracy of Doppler versus CN0.