# Interferometric DEM-Assisted High Precision Imaging Method for ArcSAR
## 2 The Geometric Model and Signal Model of ArcSAR  
Figure 1 shows the geometric model of ArcSAR system. The antenna of ArcSAR mounted by the boom rotates counterclockwise. It transmits and receives signals at equal intervals. The coordinate \(z\)-axis is the rotation axis of the ArcSAR system. We select \(xoy\) plane as the rotation plane. Point \(\circ\) is the rotation center. Point \(\circ\) represents the target in the scenes. Its coordinates can be expressed as:  
\[\left(\sqrt{R_{0}^{2}-h^{2}}\cos\varphi,\sqrt{R_{0}^{2}-h^{2}}\sin\varphi,h\right) \tag{1}\]  
where \(R_{0}\) is the distance between the target P and the point \(\circ\). \(h\) is the height of the target P. \(\varphi\) stands for the azimuth angle of the target position. Point \(\mathbb{S}_{0}\) is the position of the antenna phase center (APC). Its coordinates are  
\[\left(r\cos\theta,r\sin\theta,h\right) \tag{2}\]  
where \(r\) represents the length of the boom and \(\theta\) is the rotation angle of the boom.  
Taking example for linear frequency modulation (LFM) signal, the echo signal of the target P in ArcSAR system is shown as:  
\[\begin{split} S(\theta,t_{\rm r})=\delta_{\rm p}rect\Big{(}& \left(t_{\rm r}-2R_{\rm p}/c\right)/T_{\rm p}\Big{)}\cdot\mathrm{rect}(( \theta-\varphi)/\theta_{\rm bw})\\ &\exp\!\left(\mathrm{j}\pi K_{\rm r}\!\left(t_{\rm r}-2R_{\rm p}/ c\right)^{2}\right)\cdot\exp\!\left(-\mathrm{j}4\pi f_{\rm c}R_{\rm p}/c\right) \end{split} \tag{3}\]where \(\delta_{\rm p}\) is the backscattering coefficient, \(t_{\rm r}\) is the fast time of the ArcSAR system, \(c\) represents the speed of light, \(T_{\rm p}\) stands for the signal pulse width, \(\theta_{\rm bw}\) is the antenna beam width, \(K_{\rm r}\) represents the linear frequency modulation (LFM) rate, and \(f_{\rm c}\) is the center frequency. \(R_{\rm p}\) represents the distance between target P and the APC, which can be expressed as:  
\[R_{\rm p}=\sqrt{R_{0}^{2}+r^{2}-2r\sqrt{R_{0}^{2}-h^{2}}\cos(\theta-\varphi)} \tag{4}\]  
We perform pulse compression of \(S(\theta\),\(t_{\rm r})\) in the frequency domain on the range direction. The result can be expressed as:  
\[S(\theta,f)=\delta_{\rm p}\cdot\rm{rect}((\theta-\varphi)/\theta_{\rm bw}) \cdot\rm{rect}(f/B_{\rm r})\cdot\rm{exp}[-j4\pi(f+f_{\rm c})R_{\rm p}/c] \tag{5}\]  
where \(f\) is the current frequency in frequency domain after the spectral transformation of signal \(S(\theta\),\(t_{\rm r})\), \(B_{\rm r}\) is the bandwidth. Since the echo signal has a shift-invariant characteristic in the azimuth direction, targets with different \(\varphi\) have the same form of range migration if they are the same distance from the rotation center. Therefore, for the convenience of derivation, in this paper, we assume \(\varphi=0\). The backscattering coefficient is also neglected. \(R_{\rm p}\) and \(S(\theta,f)\) can be rewritten as a current frequency in a frequency domain:  
\[R_{\rm p}=\sqrt{R_{0}^{2}+r^{2}-2r\sqrt{R_{0}^{2}-h^{2}}\cos\theta} \tag{6}\]  
\[S(\theta,f)=\rm{rect}(\theta/\theta_{\rm bw})\cdot\rm{rect}(f/B_{\rm r}) \cdot\rm{exp}[-j4\pi(f+f_{\rm c})R_{\rm p}/c] \tag{7}\]  
Figure 1: _Cont._