# An Efficient Imaging Method for Medium-Earth-Orbit Multichannel SAR-GMTI Systems
## 3 Proposed Imaging Method
### Range Cell Migration Correction  
In order to improve the efficiency, a phase multiplication in 2-D frequency domain is used to accomplish RCMC. Based on (31), the RCM of target can be corrected by compensating the term containing \(f_{r}\). Therefore, the RCMC filter can be constructed as follows  
\[H_{RCMC}(f_{r},f_{a})=\exp\left\{\frac{j\lambda\pi f_{r}}{f_{c}}\left(\frac{1 }{4l_{20}}f_{a}{}^{2}+\frac{3l_{10}l_{30}}{8l_{20}^{3}}f_{a}{}^{2}+\frac{ \lambda l_{30}}{8l_{20}^{3}}f_{a}{}^{3}\right)\right\}. \tag{32}\]  
By multiplying (32) into (31) and then performing inverse FT (IFT), one can obtain the range-Doppler domain signal as follows:  
\[\begin{array}{ll}S_{1,trunc}(t_{r},f_{a})&=W_{a}(f_{a})p_{r}\left\{\begin{array} []{l}t_{r}-\frac{2}{2}\left(R_{0}-\frac{l_{1}^{2}}{4l_{2}^{3}}-\frac{l_{13}^{ 2}}{8l_{2}^{3}}\right)+\frac{cf_{2}^{2}}{f_{c}^{2}}\left(\frac{3l_{10}l_{30}} {16l_{20}^{3}}-\frac{3l_{13}}{16l_{2}^{3}}\right)+\frac{cf_{2}^{2}f_{c}^{2}}{f _{c}^{2}}\left(\frac{3l_{10}l_{30}}{16l_{20}^{3}}-\frac{l_{3}}{16l_{2}^{3}} \right)\\ \left(\begin{array}{c}\frac{1}{f_{c}^{2}}\frac{1}{8l_{20}}(2f_{a}\cdot M \cdot PRF+M^{2}\cdot PRF)^{2}+\frac{3l_{13}}{16l_{20}^{3}}(2f_{a}\cdot M\cdot PRF +M^{2}\cdot PRF)\\ -\frac{1}{f_{c}^{2}}\frac{1}{8l_{20}}(M^{3}\cdot PRF^{3}+3f_{a}{}^{2}M\cdot PRF +3f_{a}\cdot M^{2}\cdot PRF^{2})\end{array}\right)\end{array}\right\}\\ &\times\exp\left\{j\biggl{(}-\frac{4\pi R_{0}}{\lambda L}+\frac{\pi l_{1}}{4l _{2}^{3}}+\frac{\pi l_{13}^{2}}{2l_{20}^{3}}\biggr{)}\exp\biggl{\{}j\biggl{(} \frac{\pi l_{1}}{l_{2}}(f_{a}+M\cdot PRF)+\frac{3\pi l_{13}^{2}}{4l_{2}^{3}}( f_{a}+M\cdot PRF)\biggr{)}\biggr{\}}\\ &\times\exp\left\{j\biggl{(}\frac{\pi l_{1}}{4l_{2}}(f_{a}+M\cdot PRF)^{2}+ \frac{3\pi l_{13}l_{13}}{8l_{20}^{3}}(f_{a}+M\cdot PRF)^{2}+\frac{\pi l_{13}l _{20}^{3}}{16l_{20}^{3}}(f_{a}+M\cdot PRF)^{3}\biggr{)}\biggr{\}}.\end{array}\right. \end{array} \tag{33}\]  
From the range envelope of (33), one can see that there is still a residual RCM, which can be expressed as  
\[\Delta RCM=\left|\begin{array}{l}\frac{\lambda^{2}f_{a}^{2}}{16}\left(\frac {1}{10}-\frac{1}{l_{2}}\right)+\frac{\lambda^{2}f_{a}^{2}}{16}\left(\frac{3l_{ 10}l_{30}}{2l_{20}^{3}}-\frac{3l_{13}}{2l_{2}^{3}}\right)+\frac{\lambda^{3}f _{a}^{3}}{16}\left(\frac{l_{10}}{2l_{20}^{3}}-\frac{l_{3}}{2l_{20}^{3}}\right) \\ -\left(\begin{array}{c}\frac{3}{16l_{20}^{2}}(2f_{a}\cdot M\cdot PRF+M^{2} \cdot PRF^{2})+\frac{3\lambda^{2}l_{13}}{32l_{20}^{3}}(2f_{a}\cdot M\cdot PRF+ M^{2}\cdot PRF^{2})\\ +\frac{\lambda^{3}f_{3}}{32l_{3}^{3}}(M^{3}\cdot PRF^{3}+3f_{a}{}^{2}M\cdot PRF +3f_{a}\cdot M^{2}\cdot PRF^{2})\end{array}\right)\end{array}\right|.\end{array}\right. \tag{34}\]  
Considering \(-PRF/2<f_{a}<PRF/2\), the maximum possible value of the residual RCM is given by  
\[\Delta RCM_{\max}=\left|\Delta RCM\right|_{f_{a}=PRF/2}. \tag{35}\]  
For the targets moving with such a radial velocity that there is no Doppler ambiguity (i.e., \(M=0\)), \(\Delta RCM_{\max}\) will be smaller than half of a range resolution cell, and thus the residual RCM can be neglected. To validate this statement, numerical simulations are conducted. The MEO SAR parameters for simulations are presented in Table 1. The result is illustrated in Figure 3, which shows the \(\Delta RCM_{\max}\) of the targets with all possible velocities (note that, when there is no Doppler ambiguity, \(v_{tr}\) will be less than about 10 m/s). As can be seen from Figure 3, the maximum value of \(\Delta RCM_{\max}\) is about 1.8 m. Therefore, \(\Delta RCM_{\max}\) will not exceed half the range resolution cell in the cases that the range resolution is not high.  
The RCMC filter shown in (32) is constructed without any parameter of the target. Therefore, this filter can simultaneously correct the RCMs of static scenes and the multiple moving targets of no Doppler ambiguity. Note that after the RCM is corrected there will be no defocusing in the range direction after azimuth compression.