## 2 Materials and Methods
Assume that the beam pointing error is \(T\) times the mainlobe width \(\theta_{3 dB}\), then the corresponding coherent integration efficiency can be expressed as  
\[\widetilde{s}_{k}(t)=u_{1}(t-\Delta{t_{k}}+\overline{\tau}_{k})\mathrm{e}^{ \mathrm{i}2\pi({f_{0}}+\Delta{f_{k}})(t+\overline{\tau}_{k})}\mathrm{e}^{ \mathrm{i}\Delta{f_{k}}} \tag{22}\]5  
2 High-speed motion characteristic. Assume that \(CPI\) stands for the coherent processing interval, the tangential velocity of the target relative to the satellite is \(V_{re}\), and \(R_{0}\) denotes the radar detection range. Thus, for one \(CPI\), the maximum viewing angle offset caused by the high-speed motion characteristic is given by  
\[\widetilde{s}_{k}(t)=u_{1}(t-\Delta{t_{k}}+\overline{\tau}_{k})\mathrm{e}^{ \mathrm{i}2\pi({f_{0}}+\Delta{f_{k}})(t+\overline{\tau}_{k})}\mathrm{e}^{ \mathrm{i}\Delta{f_{k}}} \tag{22}\]6  
where atand denotes arctangent operation. Let the normalized antenna pattern and the antenna beam directivity be \(F(\theta)\) and \(\theta_{0}\), and the corresponding coherent integration efficiency can be shown by  
\[\widetilde{s}_{k}(t)=u_{1}(t-\Delta{t_{k}}+\overline{\tau}_{k})\mathrm{e}^{ \mathrm{i}2\pi({f_{0}}+\Delta{f_{k}})(t+\overline{\tau}_{k})}\mathrm{e}^{ \mathrm{i}\Delta{f_{k}}} \tag{22}\]7Based on the above-mentioned analysis, the final coherent integration efficiency of the distributed satellite constellation is given by  
\[\widetilde{s}_{k}(t)=u_{1}(t-\Delta{t_{k}}+\overline{\tau}_{k})\mathrm{e}^{ \mathrm{i}2\pi({f_{0}}+\Delta{f_{k}})(t+\overline{\tau}_{k})}\mathrm{e}^{ \mathrm{i}\Delta{f_{k}}} \tag{22}\]8  
Step 3 Multi-dimensional low sidelobe beamforming for the distributed coherent radar. For the multi-dimensional low sidelobe beamforming method, each satellite in the distributed constellation synchronously sends pulse signals with the same pulse repetition interval (PRI), where the frequency points within different \(CPIs\) are changeable. We assume that \(f_{1},f_{2},\cdots,f_{K}\) denote the \(K\) frequency points, and the corresponding beam-patterns are \(F_{1}(\theta),F_{2}(\theta),\cdots,F_{K}(\theta)\). The multi-frequency pattern \(\widehat{F}(\theta)\) can be obtained by multiplying the absolute value of these pattern functions related to different frequency points, i.e., \(\widehat{F}(\theta)=\sqrt{\prod\limits_{j=1}^{K}\left|F_{j}(\theta)\right|}\). Based on the high-speed motion characteristics of satellite constellation, the multi-frequency pattern corresponding to \(L\) sets of different positions can be generated by repeating the above- mentioned steps. On the basis of multiplying the absolute value of the acquired multi-frequency pattern, the final synthetic beampattern can be written as \(\overline{F}(\theta)=\sqrt{\prod\limits_{t=1}^{L}\left|\widehat{F}_{t}( \theta)\right|}\). In this way, the multi-dimensional joint low sidelobe beamforming for distributed satellite constellation can be achieved. Based on the above statement, a genetic algorithm (GA) [30] is adopted to design low sidelobe beamforming. Here, the satellite positions \([\{x_{i}y_{i}\}],1\leqslant i\leqslant N\), as well as the radar frequency points \(\{f_{j}\},1\leqslant j\leqslant K\), are optimization variables to achieve the minimum peak-to-sidelobe ratio (PSLR) of the distributed satellite constellation, where these variables are regarded as individuals in GA. For each individual, in order to achieve high calculating accuracy and low computation complexity, genes are constructed by real floating point numbers, i.e., a real value encoding method. As a result, not only is better population diversity in GA mutation process maintained, but the shortcoming of the poor operating performance of binary coding can be relieved. Furthermore, the distance of adjacent satellites should be greater than the minimum safe distance \(d_{c}\), and the optimized frequency points should be selected from the L-band (1 GHz\(\simeq\)2 GHz). We assume \(V_{re}\) is the radial velocity between the target and the satellite.