# An Efficient Imaging Method for Medium-Earth-Orbit Multichannel SAR-GMTI Systems
## 3 Proposed Imaging Method
### Azimuth Compression  
To finely focus the static scene, we perform the azimuth compression in the range Doppler domain. Based on (33), the azimuth compression filter is constructed as follows  
\[H_{ac}(f_{a})=\exp\Biggl{\{}-j\Biggl{(}\pi\lambda\frac{1}{4l_{20}}f_{a}{}^{2}+ \pi\lambda\frac{3l_{10}l_{30}}{8l_{20}^{3}}f_{a}{}^{2}+\frac{\pi l_{30}\lambda^ {2}}{16l_{20}^{3}}f_{a}{}^{3}\Biggr{)}\Biggr{\}}. \tag{36}\]  
The target signal after azimuth compression is given by (the negligible residual RCM is omitted)  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Orbit’s height & 6000 km \\ \hline Range bandwidth & 30 MHz \\ \hline Sampling frequency & 40 MHz \\ \hline Azimuth resolution & 6 m \\ \hline Illumination time & 3.3 s \\ \hline \(R_{e}\) & 6371 km \\ \hline \(R_{0}\) & 7548.7 km \\ \hline _PRF_ & 1400 Hz \\ \hline \(\Omega\) & 0 \\ \hline \(w\) & 0 \\ \hline \(e\) & 0 \\ \hline \(\Omega_{\rm{C0}}\) & 0 \\ \hline \(\theta_{i}\) & 90\({}^{\circ}\) \\ \hline \(\lambda\) & 0.03 m \\ \hline \(d\) & 2 m \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of MEO SAR.  
Figure 3: Residual RCM of the target with all possible velocities (i.e., \(v_{tr}\) and \(v_{th}\)). The color indicates the value of \(\Delta RCM_{\rm{max}}\), and the unit is \(m\). The distance between the target and the scene center is set to be 7568.7 km. The accelerations are 0.1 m/s\({}^{2}\).  
\[S_{1,\text{ac}}(t_{r},f_{a}) =W_{a}(f_{a})p_{r}\left\{\begin{array}{l}t_{r}-\frac{c}{2}\left(R_{0 }-\frac{d_{1}^{2}}{4k_{2}}-\frac{d_{10}^{2}}{8k_{2}^{2}}\right)-\frac{c}{f_{a}^ {2}}\frac{1}{8k_{2}^{2}}\left(2f_{a}\cdot M\cdot PRF+M^{2}\cdot PRF^{2}\right)\\ -\frac{c}{f_{a}^{2}}\frac{3(l_{1})}{16k_{2}^{2}}\left(2f_{a}\cdot M\cdot PRF+M^ {2}\cdot PRF^{2}\right)-\frac{c^{2}}{f_{a}^{2}}\frac{l_{10}}{16k_{2}^{2}}\left( M^{3}\cdot PRF^{3}+3f_{a}^{2}M\cdot PRF+3f_{a}\cdot M^{2}\cdot PRF^{2}\right) \end{array}\right\} \tag{37}\] \[\times\exp\left\{i\pi\left(-\frac{4R_{0}}{4k_{2}}+\frac{d_{1}^{2} }{4k_{2}^{2}}+\frac{d_{10}^{2}}{24k_{2}^{2}}\right)\exp\left\{i\pi\cdot M \cdot PRF\cdot\left(\frac{1}{2}+\frac{3d_{10}^{2}}{4k_{2}^{2}}+\frac{d_{2}}{4 k_{2}}\cdot M\cdot PRF+\lambda\frac{3l_{11}}{8k_{2}^{2}}\cdot M\cdot PRF+\frac{l_{1 3}l_{2}^{2}}{16k_{2}^{2}}\cdot M^{2}\cdot PRF^{2}\right)\right\}\right\}\] \[\times\exp\left\{i\pi\left(\frac{l_{1}}{k_{2}}+\frac{3d_{10}^{2}}{4 k_{2}}\right)f_{a}\exp\left\{i\pi\cdot M\cdot PRF\left(\frac{\lambda}{2}+ \frac{3l_{13}}{4k_{2}^{2}}+\frac{3l_{13}}{16k_{2}^{2}}\cdot M\cdot PRF\right) f_{a}\right\}\right.\] \[\times\exp\left\{i\pi\lambda\left[\left(\frac{l_{1}}{k_{2}}- \frac{1}{8k_{2}}\right)f_{2}+\left(\frac{3l_{13}}{8k_{2}^{2}}-\frac{3l_{10}y_{ 0}}{8k_{20}^{2}}\right)f_{a}^{2}+\left(\frac{3l_{13}}{16k_{2}^{2}}-\frac{3l_{ 20}}{16k_{20}^{2}}\right)f_{a}^{2}\right]\right\}\exp\left\{i\pi\frac{3l_{13} }{16k_{2}^{2}}\cdot M\cdot PRF\cdot f_{a}^{2}\right\}.\]  
The focused SAR image can be obtained via applying an azimuth IFT to (37). From (37) it can be seen that, for static targets, they can be finely focused. For moving targets, since there are residual azimuth modulations, they will suffer from azimuth defocusing. It should be noted that, since the RCMs of the moving targets without Doppler ambiguity have been corrected, the range defocusing will not occur after azimuth compression. Therefore, the SCNR of the target can be improved, which will improve the GMTI performance of the system.