## 2 Materials and Methods
The multi-dimensional low sidelobe beamforming problem can be formulated as  
\[ \begin{array}{ll}\min&\text{PSLR}\left(\begin{array}{c}[x_{1}y_{1}],[x_{2} y_{2}],\cdots,[x_{N}y_{N}],\\ f_{1},f_{2},\cdots,f_{K},\widehat{F}_{1},\widehat{F}_{2},\cdots,\widehat{F}_{L} \end{array}\right)\\ \text{s.t.}&d_{c}\leq\sqrt{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j} \right)^{2}},\quad 1\leqslant j<i\leqslant N\\ &\left|x_{i}-x_{j}\right|\leq D,\quad\left|y_{i}-y_{j}\right|\leq D,\quad 1 \leqslant j<i\leqslant N\\ &1e^{\theta}<f_{ii}<2e^{\theta},\quad 1\leqslant ii\leqslant K\\ &L\leq\frac{\xi}{4\text{BS}V_{re}CT}\end{array}  \tag{40}\]  
In order to evaluate each individual when one employs the GA method, the fitness function is constructed according to the maximum PSLR, shown by  
\[fit\left( \begin{array}{c}[x_{1}y_{1}],[x_{2}y_{2}],\cdots,[x_{N}y_{N}],\\ f_{1},f_{2},\cdots,f_{K},\widehat{F}_{1},\widehat{F}_{2},\cdots,\widehat{F}_{L} \end{array} \right)=\max\left\{\left|\frac{\overline{F}(u)}{\overline{F}_{\max }}\right|\right\} \tag{41}\]  
where \(\overline{F}_{\max}\) stands for the peak value of the mainlobe pattern. \(u\) represents the sidelobe region. By searching two bilateral valleys closest to the mainlobe, the desired sidelobe region could be calculated from these valleys to the direction away from the mainlobe. The optimization objective function can be described as  
\[ \begin{array}{l}\min\biggl{\{}fit\biggl{(}\begin{array}{c}\left[x_{1}\,y_{1} \right],\left[x_{2}\,y_{2}\right],\cdots,\left[x_{N}\,y_{N}\right],\\ f_{1},f_{2},\cdots,f_{K},\widehat{F}_{1},\widehat{F}_{2},\cdots,\widehat{F}_{L} \end{array}\biggr{)}\biggr{\}}\\ =\min\biggl{\{}\max\biggl{\{}\left|\frac{\widehat{F}(u)}{\widehat{F}_{\max}} \right|\biggr{\}}\biggr{\}}\end{array}  \tag{42}\]  
In order to demonstrate the optimization technique clearly, a pseudo code is presented in Algorithm 1. ``` \(\left[\left\{\mathrm{PSLR}_{g}\right\}_{g=1}^{Generations}\right]=\mathrm{GA} \left[\left\{\begin{array}{c}\mathrm{PSLR}_{g-1},\left[x_{1}\,y_{1}\right], \left[x_{2}\,y_{2}\right],\cdots,\left[x_{N}\,y_{N}\right],\\ f_{1



## 3 Results

We assume the constellation configuration is a 6\(\times\)6 square array with an anti-collision distance of 50 m, as shown in Figure 4. The beam pointing error is set to one percent of the mainlobe width. The considered simulation parameters are summarized in Table 1. The constellation height is set to 500 km, and the reason is that low orbit is conductive to decreasing the radar detection range. However, if the satellite's height is too low, it will suffer from atmospheric resistance, which will reduce the satellite's lifetime. As a result, 500 km is a suitable value. The signal bandwidth, coherent processing interval, and target RCS are selected according to the target motion characteristics and its scattering characteristics. The peak transmission power and antenna gain are calculated based on the radar detection link. When the radar system loss and noise coefficient are selected on the basis of L-band radar system composition, the total radar signal processing gain is 86.9 dB. Figure 4: Distributed satellite constellation configuration generated by STK software. Figure 3: Flowchart of dynamic coherent radar designing method. In order to implement the proposed distributed satellite constellation system, a circular relative orbit configuration is used. As the circular orbit configuration diverges [31; 32] slowly over time, the long-term stable configuration has been designed to maintain the fixed relative geometry based on the J2 perturbation correction strategy. Meanwhile, the distributed satellite configuration will also be affected by atmospheric drag, third-body gravitation, and other perturbation forces, so we employ the configuration maintenance method based on relative distance measurement and Gaussian perturbation equation to realize a highly stable orbit, in which the fuel consumption for correcting the same position accuracy can be reduced by adopting the appropriate control theory.