## 2 Materials and Methods
Therefore, it is necessary to determine the effective detection range before designing the distributed coherent radar. (1) Coherent detection constraint. Owing to the fact that a large distance exists between adjacent satellites, the coherent integration performance will be affected to some extent when the detecting target is within a certain size. In this section, the coherent integration efficiency is quantitatively analyzed based on the literature [29]. For extended targets with the size \(\Delta X\times\Delta Y\) along the plane X-Y, the geometric center is located at \(\mathbf{p}=\left(p_{x},p_{y},p_{z}\right)^{\mathsf{T}}\). After compensating for the time delay from the radar to the target geometric center, the echo signal corresponding to the transmitter \(k\)/receiver \(l\) pair \((k,l)\) can be expressed as  
\[\widetilde{r}_{k,l}(t)=\rho_{k,l}u(t) \tag{5}\]  
Figure 2: The target searching diagram of coherent radar in the whole field. that is \(\left\|{{\bf{u}}_{k}-{\bf{p}}}\right\|_{2}\geq 2D^{2}/\lambda\), \(k=1,2,\cdots,K\), where \(D\) represents the array aperture. Then the effective detection area under the far-field condition is constrained as  
\[S_{F}=\left\{{{\bf{p}}\in W\left|{\min_{k=1,2,\cdots,N}\{\left\|{{\bf{u}}_{k}-{ \bf{p}}}\right\|_{2}\}\geq\frac{2D^{2}}{\lambda},\right\}}\right. \tag{16}\]  
(3) System link constraint. The lower bound of the detection range is determined by two factors, including the coherent detection constraint condition and the far-field constraint condition, whereas the upper bound is limited by the minimum detectable signal-to-noise ratio \(\left({S/N}\right)_{0}\). According to the radar equation, system link constraints of the coherent radar are given by  
\[R_{\max}=\left({\frac{{{P_{t}}{G^{2}}{\lambda^{2}}\sigma M}}{{{\left({4\pi} \right)}^{3}}{{K_{0}}}{{F}}{{B}}{{\left({S/N}\right)}_{0}}{{L}_{s}}}}\right)^{ \frac{1}{2}} \tag{17}\]  
where \({P_{t}}\) stands for the transmit peak power, \(G\) represents the antenna gain, \(\sigma\) indicates the target RCS, and \(k\) and \({T_{0}}\) indicate the Boltzmann constant and the equivalent noise temperature. \(F\) is the noise coefficient, \(B\) is the receiver bandwidth, and \({L_{s}}\) is the system loss, and hence the signal processing gain \(M\) is given by  
\[F(\theta,\phi)=F_{\rm e}(\theta,\phi)\sum_{n=0}^{N-1}\mathrm{e}^{\frac{2\pi n }{\lambda}\Delta R_{n}} \tag{2}\]0  
In Equation (18), \(\tau\) stands for the signal bandwidth and \({N_{chirp}}\) is the integration pulse number within a coherent processing cycle. \(\eta\) is the coherent integration efficiency. Combining the above constraints, the final detection range interval of the distributed coherent radar satisfies  
\[F(\theta,\phi)=F_{\rm e}(\theta,\phi)\sum_{n=0}^{N-1}\mathrm{e}^{\frac{2\pi n }{\lambda}\Delta R_{n}} \tag{2}\]1  
where the set \(S({\bf{p}})\) is represented by  
\[S({\bf{p}})=\left\{{ \begin{array}{*{20}{c}}{{R({\bf{p}})}\left|{{R({\bf{p}}) }=\begin{array}{*{20}{c}}{{\min_{{\bf{p}}\in S_{F}\cap S_{C},}}}\\ {{k=1,2,\cdots,N}}\end{array}}\end{array} }\right.\left\{{\left\|{{\bf{u}}_{k}-{ \bf{p}}}\right\|_{2}}\right\}}\\ \end{array}}\right\} \tag{20}\]  
In particular, if the distributed satellite constellation is limited to a certain size and all satellites operate with approximately the same viewing angle, Equation (19) can be simplified as  
\[\frac{2D^{2}}{\lambda}\leq R\leq{R_{\max}} \tag{21}\]  
Step 2 Coherent integration efficiency analysis of distributed coherent radar. For a dynamic coherent radar, the coherent integration efficiency is mainly influenced by several factors, i.e., system synchronization error, beam pointing error, and high-speed motion characteristics. Here, a quantitative evaluation of these items is carried out. (a) The impact of system synchronization error on coherent integration efficiency. The considered system synchronization errors are composed of position, time, frequency, and phase synchronization errors.