## 2 Materials and Methods
## 4 Discussion

In this paper, the impacts of synchronization error, beam pointing error, and high-speed motion characteristics on coherent integration have been systematically evaluated in our studies for a spaceborne satellite constellation. On this basis, the angle ambiguity problem is discussed in consideration of the distributed dynamic coherent radar configuration. From Section 2, the target should be in the far-field for a distributed satellite constellation when conducting a coherent detection process. As a result, the constellation scale is limited, and low sidelobe beamforming cannot be performed in the spatial domain with limited DOFs in a robust way. In order to enhance the unambiguous field, the multi-dimensional joint method, involving three spatial DOFs, multi-carrier-frequency, and temporal virtual array elements, is adopted to suppress the grating lobes and achieve robust low-sidelobe beamforming. The strategy for reducing grating lobes and sidelobes can be summarized as follows: Firstly, high-speed motion characteristics are employed to form virtual apertures within different \(CPIs\), and hence increase the spatial domain DOFs. More

Figure 11: The beampattern under elevation angle of 45 and azimuth angle of 135. (**a**) The elevation pattern; (**b**) the azimuth pattern. Figure 10: The beampattern under elevation angle of 30 and azimuth angle of 120. (**a**) The elevation pattern; (**b**) the azimuth pattern. over, multiple frequency points are introduced in the frequency domain to further improve the radar system DOFs. Finally, the non-uniform array optimization criterion is adopted to minimize the system's grating lobes and sidelobes, where safe distances between satellites are guaranteed. This strategy has two advantages: On one hand, the integration time within one \(CPI\) is at approximately the millisecond level, and consequently, temporal virtual array elements are obtained by utilizing high-speed motion characteristics of the satellite so that the blank space between adjacent platforms can be compensated. On the other hand, the multi-carrier-frequency method is introduced based on spatial and temporal beamforming, that is, the multi-frequency beampatterns are combined to obtain low sidelobe beamforming. ## 5 Conclusions

The distributed coherent radar array achieves wide applications in the space surveillance field. However, the dense grating lobe problem inevitably exists when one conducts radar array beamforming of a sparse satellite configuration. From the perspective of distributed radar coherent detection constraints and the robust low-sidelobe beamforming technique, a novel transmit beampattern optimization method for dynamic coherent radar based on a distributed satellite constellation is presented, which provides a feasible coherent detection scheme for the distributed satellite constellation. It should be noted that the mainlobe for the distributed array pattern is extremely narrow, which seriously contradicts the data rate. As a result, the trade-off between the dense grating lobes and data rate should be balanced. Conceptualization, X.T., C.D. and Y.L. ; methodology, X.T. and C.D. ; software, X.T., Y.L., J.C. and J.A. ; validation, J.C. and J.A. ; formal analysis, X.T. ; investigation, Y.L. and J.C.; resources, X.T. and C.D. ; data curation, C.D. and Y.L. ; writing--original draft preparation, X.T. ; writing--review and editing, Y.L. and J.C.; visualization, Y.L. and J.C.; supervision, J.C. and J.A. ; project administration, Y.L. and J.C.; funding acquisition, Y.L. All authors have read and agreed to the published version of the manuscript. This research was funded by National Natural Science Foundation of China under grant number 62107033, National Defense Science and Technology Foundation Strengthening Plan Fund under grant number 2022-JCJC-JJ-0202, and the Outstanding Youth Foundation of the Xi'an branch, the Fifth Academy, Aerospace Science, and Technology Group under Grant Y21-RCFYQ1-05. Not applicable.