# LEO-Augmented GNSS Based on Communication Navigation Integrated Signal
## 2 Performance of the Navigation Signal
### Pseudorange 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 pseudorange.  
The pseudorange accuracy of open loop estimation based on the noncoherent early minus later power discriminator is shown below [16].  
\[\sigma_{OL}=\sqrt{\frac{\tau_{b}}{2C/N_{0}}\frac{1}{P_{\ell}T_{\rm coh}}\bigg{(} 1+\frac{1}{C/N_{0}\cdot T_{\rm coh}(1-\tau_{b})}\bigg{)}}(\textit{chips}) \tag{6}\]  
where \(\tau_{b}\) is the search interval, \(T_{coh}\) is the coherent integration time, and \(C/N_{0}\) is the CN0 of the navigation signal.  
The pseudorange accuracy caused by thermal noise for DLL based on the noncoherent early minus later power discriminator is shown below when the correlation interval D is set to 0.5 [14].  
\[\sigma_{DLL}=\sqrt{\frac{B_{L}}{2\cdot C/N_{0}}\frac{1}{B_{\ell}T_{c}}\bigg{(} 1+\frac{1}{T_{coh}C/N_{0}}\bigg{)}}(\textit{chips}) \tag{7}\]  
where \(B_{L}\) is the loop noise bandwidth and it is set as 1 Hz in this paper, \(B_{\ell}\) is the radio frequency bandwidth, and \(T_{c}\) is the chip width. In this paper, \(B_{\ell}\) is assumed to be \(2f_{c}\), and \(f_{c}\) is the code rate.  
Considering that, for burst signal, the receiver does not have to deal with the signal in real-time, the correlation interval can be lower and \(T_{coh}\) can be longer. Therefore, \(\tau_{b}\) is set as 0.1, which is lower than the correlation interval D of DLL.  
Figure 3: The PSD of signal by generated method 3.  
In the following the pseudorange accuracy of open loop estimation and DLL are compared. The following parameters are set to get the comparison result.  
\(\sigma_{OL}^{1}\): \(B_{nav}=31.5\) KHz, \(T_{coh}=1\) ms \(\sigma_{OL}^{2}\): \(B_{nav}=31.5\) KHz, \(T_{coh}=90\) ms \(\sigma_{\rm DLL}^{1}\): \(B_{nav}=10.5\) MHz, \(T_{coh}=1\) ms \(\sigma_{\rm DLL}^{2}\): \(B_{nav}=31.5\) KHz, \(T_{coh}=1\) ms  
\(\sigma_{OL}^{1}\) and \(\sigma_{OL}^{2}\) correspond to method 2, \(\sigma_{\rm DLL}^{1}\) corresponds to method 1, and \(\sigma_{\rm DLL}^{2}\) corresponds to method 3. The following figure shows the comparison results.  
From Figure 4, we draw the following conclusions.  
1. Comparing \(\sigma_{OL}^{1}\) with \(\sigma_{OL}^{2}\) and \(\sigma_{\rm DLL}^{2}\), it can be seen that with the increment of \(T_{coh}\), \(\sigma\) decreases for open loop estimation. However, the advantage of open loop estimation over DLL is low when the bandwidth is the same.
2. Comparing \(\sigma_{OL}^{1}\) with \(\sigma_{\rm DLL}^{2}\), it can be seen that the accuracy of DLL is better than open loop estimation even though \(\tau_{b}\) is lower than D.
3. Comparing \(\sigma_{\rm DLL}^{1}\) with \(\sigma_{\rm DLL}^{2}\), it can be seen that the increment of \(B_{nav}\) causes great decrement of \(\sigma\), and, when CN0 is 77 dBHz, \(\sigma_{\rm DLL}^{2}\) is about the same with \(\sigma_{\rm DLL}^{1}\) when CN0 is 30 dBHz.  
Based on the analysis above, it can be concluded that, the accuracy of method 2 is far worse than methods 1 and 3, and \(B_{nav}\) should be increased to get higher accuracy for all methods.