## 3 Theoretical Analysis
### Imaging Characteristic between MBS and EGT  
Section 3.1 analyzes the range characteristic of MBS at four moments during a specific period. After the real trajectory has been fitted under suitable parameters, the FBP algorithm can be used for final image. Because the load on TerraSAR-X, ground-based imaging radar at Kashi, China, etc., are all X-band continuous waves, high-frequency bands have higher transmission power, and are capable of reaching kilowatts that can compensate for power loss during transmission; a band of high frequency will be less affected by the ionosphere than will the low-frequency P-band [21]. Along With increasing synthetic aperture time, the background ionosphere changes in temporal and spatial domains will also cause more serious azimuth shift and image defocusing for low-frequency bands. Here, the antenna aperture in the range or azimuth direction is up to tens of meters, for a better view in the long Earth-Moon distance; as shown in Table 7, the range width angle with only 0.028\({}^{\circ}\) still shows about a 416 km swath in range when MBS is at apogee. Among periods of the Moon's revolution, slightly considering the influence of the incidence angle and the sub-point on echo, the look angle is selected as a middle number, i.e., 0.3\({}^{\circ}\). Since the EGT's image of MBS does not require high resolution when considering a large coverage, the range bandwidth can be set to a relatively low value. The basic system parameters of MBS are shown in Table 7, and the processed nine points image of MBS's echo at four moments corresponding to the apogee, perigee, ascending, and descending nodes are expressed in Figure 6. While the final compressed image \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Apogee** & **Perigee** & **Ascending Node** & **Descending Node** \\ \hline MBS’s speed (m/s) & 971.7 & 1102.2 & 1039.8 & 997.2 \\ East’s speed (m/s) & \(-\)420.865 & \(-\)412.103 & \(-\)448.733 & \(-\)450.012 \\ North’s speed (m/s) & 2.704 & \(-\)2.184 & \(-\)6.803 & 6.265 \\ Up’s speed (m/s) & \(-\)0.017 & \(-\)0.002 & 1.168 & \(-\)1.027 \\ Distance of moon-earth centroid (km) & 4.0656 \(\times\) 10\({}^{8}\) m & 3.5713 \(\times\) 10\({}^{8}\) m & 3.7670 \(\times\) 10\({}^{8}\) m & 3.9715 \(\times\) 10\({}^{8}\) m \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The speed of MBS in ECR, the distance of Moon-Earth centroid, and the speed of corresponding sub-point in ENU orientation. The line parallel to latitude and direct to east is called “E” orientation; the line parallel to longitude and direct to north is called “N” orientation; and the other line that satisfies the right-hand rule with the “E” and “N” is called “U” orientation. The time of apogee, perigee, ascending, and descending nodes are consistent with the time in Tables 2 and 3. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Carrier frequency & 9 GHz \\ Pulse width & 100 us \\ Range bandwidth & 3 MHz \\ Azimuth oversampling & 1.2 \\ Range oversampling & 1.2 \\ Azimuth width angle & 0.014\({}^{\circ}\) \\ Range width angle & 0.028\({}^{\circ}\) \\ Look angle & 0.3\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Basic system parameters of MBS. shows little difference at these four moments, the target echo is still validly compressed using the proposed FBP algorithm. Among the simulation results, the Peak Side Lobe Ratio (PSLR), in row slice and column slice, both exceed \(-13\) dB when MBS is at apogee and perigee, but are shown as somewhat smaller when the MBS is at the ascending node and the descending node, indicating that the images at apogee and perigee are slightly better than those for other two moments.