# Navigation in GEO, HEO, and Lunar Trajectory Using Multi-GNSS Sidelobe Signals
## 2 Observation Simulation Strategy
### Transmitter and Receiver Patterns  
The key point of observation modeling is the power of the received signal, which is mainly determined by the GNSS transmitter radiation pattern and receiver antenna gains. For GPS Blocks IIR, IIR-M, and IIF, their transmitter radiation patterns were adopted from published data by Lockheed Martin, the GPS manufacturer company, and the National Aeronautics and Space Administration (NASA) [23; 24]. We used the Block IIA radiation pattern to present the Block III-A radiation pattern since there was no available data of Block III-A yet. For BDS, there were no public antenna pattern data for B1I. To overcome this problem, the GPS Blocks IIF satellite's radiation pattern was used, and the beamwidths difference between GPS and BDS was considered according to the beamwidth values given by [13]. The same strategy was applied to GLONASS and Galileo for the same reason. Note that the published radiation pattern varies with both azimuths and off-boresight angles. To simplify the computation, we used an averaging antenna pattern over the azimuth. The symmetric GNSS equivalent isotropic radiated power (EIRP) patterns are shown in Figure 2, and the off-boresight angle was denoted as \(\theta\) (Figures 3 and 4). Note that the GNSS patterns were symmetric about the off-boresight, and signals between \(-90^{\circ}\) and \(90^{\circ}\) were all used.  
For receiver antenna gain simulation, we assumed that the spacecraft was equipped with nadir and zenith antennas that are able to track multi-GNSS signals. The receiving off-boresight angle is denoted as \(\psi\) in Figure 3[17]. When the spacecraft moved away from the GNSS satellites, the received signal powers were attenuated due to transmission loss. In this case, a high-gain antenna was mounted on the nadir-pointing receiver. When the spacecraft moved below the GNSS constellation, a zenith-pointing patch antenna ensured satellite visibility. According to [13], the receiver antenna gain of the two antennas are shown in Figure 5.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Altitudes** & **Inclination** & **RAAN (\({}^{\circ}\))** & **Argument of** & **True** \\  & **[km]** & **Eccentricity** & **(\({}^{\circ}\))** & **RAAN (\({}^{\circ}\))** & **Perigee (\({}^{\circ}\))** & **Anomaly (\({}^{\circ}\))** \\ \hline GEO & 35,786 & 0 & 0 & 0 & 0 & 0 \\ \hline \hline \multicolumn{7}{l}{* Epoch: 20 May 2021 00:00:00–21 May 2021 00:00:00 (UTC).} \\ \end{tabular}
\end{table}
Table 4: HEO Parameter *.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Semi-Major** & **Eccentricity** & **Inclination** & **RAAN** & **Argument of** & **True** \\  & **Axis (km)** & & **(\({}^{\circ}\))** & **(\({}^{\circ}\))** & **Perigee (\({}^{\circ}\))** & **Anomaly (\({}^{\circ}\))** \\ \hline Start & 67,500.44 & 0.8968 & 31 & 181 & 179.8 & 357.8 \\ Stop & 294,807.182 & 0.9944 & 43.168 & 174.6 & 175.8 & 175.94 \\ \hline \hline \multicolumn{7}{l}{* Epoch: 20 May 2021 22:48:00–24 May 2021 03:00:00 (UTC).} \\ \end{tabular}
\end{table}
Table 5: Lunar trajectory Parameter *.  
Figure 4: GNSS antenna coordinate system.  
Figure 3: Physical visibility of HEO spacecraft.  
Figure 2: GNSS symmetric antenna patterns.  
As GNSS EIRP and receiver antenna gain \(G_{r}(\psi)\) were simulated, power levels \(P_{R}\) received by spacecraft could be calculated as [34]:  
\[P_{R}=EIRP+G_{r}(\psi)+L_{S} \tag{1}\]  
where \(L_{S}=20\lg(\frac{\lambda}{4\pi d})\) is the free space propagation loss; \(\lambda\) is the wavelength, and \(d\) is the transmission distance. \(C/N_{0}\) is expressed as:  
\[C/N_{0}=P_{R}-N_{0} \tag{2}\]  
where \(N_{0}=kT\), \(k\) is the Boltzmann constant and equals 1.38 \(\times\) 10\({}^{-23}\) J/K. \(T\) is the ambient temperature, which was set to be 175.84 K. In this paper, the tracking and acquisition threshold was 20 dB-Hz. Considering the transmission loss due to long distance, 15 dB-Hz receiver threshold of lunar trajectory was also analyzed.