# Acquisition of Weak BDS Signal in the Lunar Environment
## 2 Materials and Methods
### Observation Scenarios  
GNSS satellites were originally designed to navigate for low-orbiting spacecraft and ground users, and therefore their signals are transmitted towards the Earth. Lunar spacecraft can only receive those signals that are not obscured by the Earth, most of which are emitted by the sidelobes, as seen in Figure 4. The red and green areas indicate the emitted signals from the main and side lobes, respectively. It can be seen that the receiver on the other side of the Earth relative to the GNSS satellite can only acquire a small portion of the main lobe signals. Table 2 gives the Earth blocking angle which is related to the orbital altitude. When the angle formed by the receiver, the GNSS satellite, and the geocenter is less than the blocking angle, the signal is considered unreachable to the receiver. Once a GNSS satellite is visible to a spacecraft, the received signal power, \(P_{R}\), of the lunar receiver can be expressed as [31]:  
\[P_{R}=P_{T}+G_{T}(\alpha)+G_{R}(\beta)+L_{S}-L_{A}, \tag{1}\]  
where \(P_{T}\) is the output power of the transmitter amplifier and \(G_{T}\) is the transmitter antenna gain, which is related to the off-boresight angle \(\alpha\). \(G_{R}\) is the receiver antenna gain related to off-boresight angle \(\beta\) and \(L_{S}\) is the free space propagation loss. As the GNSS signal passes through the troposphere, it covers only a small portion of the entire propagation; hence, the effect of the atmosphere is ignored and the atmospheric loss, \(L_{A}\), is set to 0 dB in this \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**System**} & \multirow{2}{*}{**GPS**} & \multicolumn{3}{c}{**BDS**} & \multirow{2}{*}{**GLONASS**} & \multirow{2}{*}{**Galileo**} \\ \cline{3-3} \cline{5-6}  & & **MEO** & & & \\ \hline Earth & & & & & \\ blocking & 13.5 & 12.9 & 8.6 & 14.1 & 12.2 \\ angle 1 (deg) & & & & & \\ \hline \hline \end{tabular}  
* The Earth blocking angle is the angle between the line connecting the GNSS satellite and the centre of the Earth and the tangent line passing through the GNSS satellite.  
\end{table} 
Table 2: Earth blocking angle for each GNSS system. Figure 4: A simulation scenario where the GNSS signal can reach an LLO satellite. \(\alpha\) and \(\beta\) are the off-boresight angles of the transmitted and received signals, respectively. The red and green areas are the main and side lobe areas, respectively. \(\theta\) indicates the maximum off-boresight angle of the main lobe signal. (Please note that the distances shown are not to scale and are intended as an illustration only). simulation. The free space propagation loss, \(L_{S}\), is determined by the propagation distance, \(d\), and the signal frequency, \(f\):  
\[L_{S}=20 lg(\frac{c}{2\pi df}), \tag{2}\]  
where \(c\) is the speed of light in a vacuum. Based on the received power, \(P_{R}\), the signal-to-noise ratio (SNR) is denoted as:  
\[SNR=P_{R}-N, \tag{3}\]  
where \(N\) is the noise power in dB, which can be written as:  
\[N=10 lg(kTB_{n}), \tag{4}\]  
where \(k=1.3806505\times 10^{-23}\) J/K is the Boltzmann constant. The noise temperature of the system \(T\) is set to a constant of 175.84 K in our simulation as this is close to the real conditions in SSV applications [5]. \(B_{n}\) is the radio frequency (RF) front-end bandwidth, and this value is closely related to the SNR. In order to compare the performance of different receivers, we usually calculate the carrier/noise ratio, \(C/N_{0}\). It can be expresed as:  
\[C/N_{0}=P_{R}-N_{0}, \tag{5}\]  
where \(N_{0}=10 lg(kT)\) is the noise power spectrum density and \(C/N_{0}\) is independent of the noise power generated by the receiver front-end bandwidth.