## 2 Materials and Methods
In this way, the vacancies between adjacent satellites are compensated to further suppress the grid lobes as well as the sidelobes. Figure 7 shows the virtual array diagram of the

Figure 6: Integration efficiency with different non-ideal factors. (**a**) Influence of time synchronization error; (**b**) influence of phase synchronization error; (**c**) influence of beam pointing error; (**d**) influence of high-speed motion characteristics. distributed satellite constellation within three \(CPIs\), including the real aperture related to the current \(CPI\) and the virtual aperture related to the other two \(CPI\)s.

For the proposed multi-dimensional low sidelobe beamforming method, based on the minimum sidelobe objective function and GA [30], the optimized frequencies \(f_{1},f_{2},f_{3},f_{4}\), \(f_{5},f_{6},f_{7},f_{8},f_{9},f_{10}\) are shown in Table 2, where the lower limit of the satellite safety distance and that of the target detection distance are set to 50 m and 600 km, respectively. Figure 8 shows the beampattern under an elevation angle of 90 and an azimuth angle of 90. One can see that the first sidelobe is approximately 2 dB if only a single frequency is applied, which cannot meet the detection requirements. Both maximum sidelobes corresponding to the elevation and azimuth dimensions are better than \(-13.5\) dB if the space-frequency-time joint optimization strategy is adopted, and it possesses a slight improvement compared to the space-frequency joint optimization algorithm. From the stochastic perspective, the target velocity is 5 km/s, and the time-sensitive target has a radial velocity relative to the satellite. Therefore, the clutters whose Doppler interval coincides with the target are generally located \(5^{\circ}\) away from the beam center, and the two-dimensional sidelobes within these regions are better than \(-30\) dB, which meets the requirement for weak target detection. In order to further verify the performance of the robust low sidelobe beamforming algorithm, the maximum sidelobe of beam patterns with regard to different antenna-pointing angles are demonstrated in Figures 9-11, which are under an elevation angle of 60 and an azimuth angle of 120, an elevation angle of 30 and an azimuth angle of 120, and an elevation angle of 45 and an azimuth angle of 135, respectively. One can see that the first sidelobes corresponding to different scanning angles are better than \(-14\) dB by means of the space-frequency-time joint optimization strategy. Furthermore, the sidelobes beyond 5 degrees are better than \(-30\) dB. Thus, the sidelobe level can be greatly reduced by utilizing the space-frequency joint optimization beamforming algorithm compared with the single-frequency beamforming algorithm and space-frequency joint beamforming algorithm. Different from the orthogonal mode, the equivalent antenna aperture becomes \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Frequency Point** & \(f_{1}\) & \(f_{2}\) & \(f_{3}\) & \(f_{4}\) & \(f_{5}\) \\ \hline
**Value(Hz)** & 1.0000 GHz & 1.0782 GHz & 1.1233 GHz & 1.1869 GHz & 1.2500 GHz \\ \hline
**Frequency Point** & \(f_{6}\) & \(f_{7}\) & \(f_{8}\) & \(f_{9}\) & \(f_{10}\) \\ \hline
**Value(Hz)** & 1.5853 GHz & 1.6491 GHz & 1.6787 GHz & 1.7962 GHz & 1.8176 GHz \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Radar system carrier frequency optimization results. Figure 7: The virtual array diagram of distributed satellite constellation within three \(CPIs\). smaller when one conducts the beam scanning procedure, which could relieve the sparsity of the satellite constellation. The disadvantage is that the mainlobe of the beampattern is widened and the gain of the antenna array is decreased. From Figures 8-11, one can see that the proposed robust low-sidelobe beamforming method, with limited spatial degrees of freedom (DOFs), can be realized under the beam scanning mode and distributed array configuration, where the coherent integration efficiency can be optimized to more than 90%.