# LEO-Augmented GNSS Based on Communication Navigation Integrated Signal
## 3 Performance of Communication Signal
### BER Performance  
To obtain the limit of the navigation signal, BER is analyzed. The procedure when receiving the communication signal is as follows. First, the signal is filtered by a band limited filter. Second, the signal is down frequency converted to zero frequency with the locally generated carrier. Third, the signal is integrated during the bit length \([0,T_{b}]\) to improve the SNR. During this process, the navigation signal in the bandwidth will also be integrated, and the power becomes jamming for the communication signal. The signal expression after the filter is shown below. \[y(t)=h(t)(S_{c}(t)+S_{nav}(t)+n(t)) \tag{13}\]  
where \(h(t)\) is the expression of the filter and the expression of \(S_{c}(t)\), shown below. \[S_{c}(t)=\sqrt{2P_{c}}b(t)\cos(2\pi(f_{0}+f_{d})t+\phi_{0}) \tag{14}\]  
where \(f_{0}\) is the default frequency of the signal, \(f_{d}\) is the Doppler frequency, \(b(t)\) is the communication bit, \(\phi_{0}\) is the initial carrier phase, and \(P_{c}\) is the power of the communication signal. After obtaining a frequency converter with local generated carrier and integration, the signal is shown below. \[ \begin{split} Y^{\prime}&=\int_{t=0}^{T_{b}}y(t) \sqrt{2}\cos(2\pi(f_{0}+f_{d})t+\phi_{0})dt\\ &=\int_{t=0}^{T_{b}}(S_{c}(t)+h(t)S_{nav}(t)+n(t))\sqrt{2}\cos(2 \pi(f_{0}+f_{d})t+\phi_{0})dt\\ &=\sqrt{P_{c}}b_{0}(0)T_{b}+\sqrt{P^{\prime}_{nav}}c^{\prime}_{nav }T_{b}+\int_{0}^{T_{b}}\sqrt{2}n(t)\cos(2\pi(f_{0}+f_{d})t+\phi_{0})dt\end{split}  \tag{15}\]  
where \(P^{\prime}_{nav}\) is the power of the navigation signal after integration and \(c^{\prime}_{nav}\) is the sign. Note that to get the worst BER performance, it is assumed, in (15), that the navigation signal will become a signal of constant power with the sign being positive or negative after the filter and integration. \(P^{\prime}_{nav}\) is equal to the accumulated power in the occupied bandwidth of the communication signal and the highest value is obtained when the communication band locates at the central of the PSD of the navigation signal. Under this condition, \(P^{\prime}_{nav}\) is expressed as  
\[P^{\prime}_{nav}=P_{nav}\frac{\int_{-B_{c}/2}^{B_{c}/2}G_{nav}(f)df}{\int_{-B_ {nav}/2}^{B_{nav}/2}G_{nav}(f)df} \tag{16}\]  
where \(B_{c}\) is the occupied bandwidth of the communication signal,\(B_{nav}\) is the bandwidth of the navigation signal, \(G_{nav}(f)\) is the PSD of the navigation signal, and \(P_{nav}\) is the power of the navigation signal. After integration, the noise is \(\Lambda\sim(0,N_{0}T_{b}/2)\). From Equation (16), it can be seen that, when the \(CNSR\) is the same, the increment of the bandwidth of the navigation signal decreases the value of \(P^{\prime}_{nav}\). Therefore, it would be an optimized option when the bandwidth of the navigation signal is as large as possible. Based on the analysis above, the BER given in [19] is as follows,  
\[P_{e}=P_{e1}+P_{e2}+P_{e3}+P_{e4} \tag{17}\]  
\[P_{e1}=P_{e4} =P_{r}\Big{[}\Lambda<\left(-\sqrt{P_{c}}T_{b}-\sqrt{P^{\prime}_{nav}}T _{b}\right)\Big{]}\] \[=\mathbb{Q}\Bigg{(}\sqrt{\frac{2E_{b}\Big{(}1+\sqrt{P^{\prime}_{nav }/P_{c}}\Big{)}}{N_{0}}}\Bigg{)} \tag{18}\]  
\[P_{e2}=P_{e3} =P_{r}\Big{[}\Lambda<\left(\sqrt{P_{c}}T_{b}-\sqrt{P^{\prime}_{nav }}T_{b}\right)\Big{]}\] \[=\mathbb{Q}\Bigg{(}\sqrt{\frac{2E_{b}\Big{(}1-\sqrt{P^{\prime}_{ nav}/P_{c}}\Big{)}}{N_{0}}}\Bigg{)} \tag{19}\]  
Considering that the data rate is 50 Kb/s, the BER performance of one unit band versus \(CN0_{total}\) is shown below when the bandwidth of the navigation signal is 10.5 MHz. This is the worst BER performance among all the communication bands. The legend of Figure 7 is the value of \(CNSR\) in dB, and 'PureCom' means that there is no navigation signal. The BER threshold is considered as \(10^{-4}\) for satellite communication, and it is shown in the figure with black full line. From Figure 7, when \(CN0_{total}\) is fixed, BER performance increases with the increment of \(CNSR\).