## 2 Materials and Methods
The transmit signal of the kth radar is established relative to the reference radar, that is,  
\[\widetilde{s}_{k}(t)=u_{1}(t-\Delta{t_{k}}+\overline{\tau}_{k})\mathrm{e}^{ \mathrm{i}2\pi({f_{0}}+\Delta{f_{k}})(t+\overline{\tau}_{k})}\mathrm{e}^{ \mathrm{i}\Delta{f_{k}}} \tag{22}\]  
where \(t\in\left[{-T_{p}}/{2,T_{p}}\left/{2}\right.\right]\) represents the signal transmit time. Let radar 1 be the reference, \(\Delta{t_{k}}\) is the time synchronization error of the kth radar, \(\overline{\tau}_{k}\) is the ideal signal propagation delay from the kth radar to the target, \(\Delta{f_{k}}\) is the frequency synchronization error of the kthradar, and \(\Delta\phi_{k}\) is composed of the channel phase error, local oscillator initial phase error, and space-time synchronization error terms, i.e., \(\Delta\phi_{k}=2\pi(f_{0}+\Delta f_{k})\Delta t_{k}+\Delta\overline{\phi}_{k}\), where \(\Delta\overline{\phi}_{k}=\Delta\overline{\phi}_{k1}+\Delta\overline{\phi}_{k2}\). After down conversion operation, the signal received by the lth radar is given by  
\[\widetilde{r}_{l}(t)=\sum_{k=1}^{N}u_{1}(t-\Delta t_{k,l}-\delta\tau_{k,l}) \mathrm{e}^{\mathrm{i}2\pi(f_{0}+\Delta f_{k})(t-\delta\tau_{k,l})}\mathrm{e}^ {\mathrm{i}\Delta\phi_{k}}\mathrm{e}^{-\mathrm{i}2\pi(f_{0}+\Delta f_{l})t} \mathrm{e}^{-\mathrm{i}\Delta\phi_{l}} \tag{23}\]  
where \(\Delta t_{k,l}=\Delta t_{k}-\Delta t_{l}\), \(\Delta\phi_{l}=2\pi(f_{0}+\Delta f_{l})\Delta t_{l}+\Delta\overline{\phi}_{l}\), with \(\Delta\phi_{l}\), in analogy to \(\Delta\phi_{k}\), involving the similar error terms. \(\delta\tau_{k,l}=\delta\tau_{k}+\delta\tau_{l}\) describes the difference between the real-time delay \(\tau_{k,l}=\tau_{k}+\tau_{l}\) and the ideal time delay \(\overline{\tau}_{k,l}=\overline{\tau}_{k}+\overline{\tau}_{l}\), with \(\tau_{k}=||\mathbf{u}_{k}+\Delta\mathbf{u}_{k}-\mathbf{p}||_{2}\) and \(\overline{\tau}_{k}=||\mathbf{u}_{k}-\mathbf{p}||_{2}\).\(\Delta\mathbf{u}_{k}\) is inevitable due to the limited measurement accuracy and uncontrollable orbit error, and we define it as the position error. Noting that the influence of the position error on the signal envelope and the synchronization error coupling term \(\Delta f_{k}\cdot\delta\tau_{k,l}\) are negligible, Equation (23) can be rewritten as  
\[r_{l}(t)\approx\sum_{k=1}^{N}\Big{[}u_{1}(t-\Delta t_{k,l})\mathrm{e}^{ \mathrm{i}2\pi\Delta f_{k}t}\Big{]}\mathrm{e}^{-\mathrm{i}2\pi f_{0}\delta \tau_{k,l}}\mathrm{e}^{\mathrm{i}\Delta\phi_{k}}\mathrm{e}^{-\mathrm{i}\Delta \phi_{l}} \tag{24}\]  
where \(\Delta f_{k,l}=\Delta f_{k}-\Delta f_{l}\). By stacking all subarray-level signals, the integrated signal can be expressed as  
\[r(t)\approx\sum_{l=1}^{N}\sum_{k=1}^{N}\Big{[}u_{1}(t-\Delta t_{k,l})\mathrm{ e}^{\mathrm{i}2\pi\Delta f_{k}t}\Big{]}\mathrm{e}^{-\mathrm{i}2\pi f_{0} \delta\tau_{k,l}}\mathrm{e}^{\mathrm{i}\Delta\phi_{k}}\mathrm{e}^{-\mathrm{i} \Delta\phi_{l}} \tag{25}\]  
The coherent integration efficiency, taking into account the system synchronization error, is represented by  
\[\eta_{1}=\frac{1}{N^{4}}E\Bigg{[}\sum_{l=1}^{N}\sum_{k=1}^{N}\sum_{l^{\prime} =1}^{N}\sum_{k^{\prime}=1}^{N}\mathrm{e}^{\mathrm{i}(\Delta\phi_{k}-\Delta \phi_{k^{\prime}})}\mathrm{e}^{\mathrm{i}(\Delta\phi_{l}-\Delta\phi_{k^{\prime }})}\mathrm{e}^{-\mathrm{i}2\pi f_{0}(\delta\tau_{k}-\delta\tau_{k^{\prime}}+ \delta\tau_{l}-\delta\tau_{l^{\prime}})}\chi\big{(}\Delta t_{k,l}-\Delta t_{k ^{\prime},l^{\prime}},\Delta f_{k,l}-\Delta f_{k^{\prime},l^{\prime}}\big{)} \Bigg{]} \tag{26}\]  
where  
\[\chi(\tau,f)\triangleq\int_{-\infty}^{\infty}u_{1}(t)u_{1}^{*}(t-\tau) \mathrm{e}^{\mathrm{i}2\pi ft}\mathrm{d}t \tag{27}\]  
is the ambiguity function of the baseband waveform.