# An Efficient Imaging Method for Medium-Earth-Orbit Multichannel SAR-GMTI Systems
## 4 Property Analysis of Moving Target in the Complex Image Domain
### Range Displacement  
The target's location in range is displaced from \(R_{0}\), which can be seen from the range envelope of (37). Thus, the target will suffer range displacement after focusing. Based on (37), the range displacement can be calculated as  
\[r_{shift}=\frac{l_{1}^{2}}{4l_{2}}+\frac{l_{1}^{3}l_{3}}{8l_{2}^{3}}-\frac{c} {2}\left(\begin{array}{c}\frac{c}{f_{a}^{2}}\frac{1}{8k_{2}}\left(2f_{a} \cdot M\cdot PRF+M^{2}\cdot PRF^{2}\right)+\frac{c}{f_{a}^{2}}\frac{3l_{13}}{16 k_{2}^{2}}\left(2f_{a}\cdot M\cdot PRF+M^{2}\cdot PRF^{2}\right)\\ +\frac{c^{2}}{f_{a}^{2}}\frac{l_{3}}{16k_{2}^{2}}\left(M^{3}\cdot PRF^{3}+3f_{a} ^{2}M\cdot PRF+3f_{a}\cdot M^{2}\cdot PRF^{2}\right)\end{array}\right). \tag{38}\]  
Since the terms containing \(l_{3}\) are smaller than a range cell, they can be neglected. Therefore, \(r_{shift}\) can be simplified as  
\[r_{shift} \approx\frac{l_{1}^{2}}{4l_{2}}-\frac{\lambda^{2}}{16l_{2}}\left(2f_ {a}\cdot M\cdot PRF+M^{2}\cdot PRF^{2}\right) \tag{39}\] \[=\frac{\lambda^{2}}{16l_{2}}f_{b}^{2},\]  
where \(f_{b}\) defines the target's baseband Doppler center frequency, and it can be expressed as \(f_{b}=-2v_{tr}/\lambda-M\cdot PRF\).  
In order to reveal the relationship between the range displacement and the target's parameters, the expressions of \(l_{2}\) and \(f_{b}\) are substituted into (39), and \(r_{shift}\) can be rewritten as  
\[r_{shift} \approx\frac{\lambda^{2}R_{0}(-2v_{tr}/\lambda-M\cdot PRF)^{2}}{8( v_{0}^{2}-v_{tr}^{2}-2v_{tr}\omega_{0}+v_{0}^{2}+a_{tr}\omega_{0}+a_{tr}R_{0}-a_{tr}R_{0})} \tag{40}\] \[\approx\frac{\lambda^{2}R_{0}(2v_{tr}/\lambda+M\cdot PRF)^{2}}{8( -2v_{tr}\omega_{0}+v_{0}^{2}+a_{tr}R_{0}-a_{tr}R_{0})}.\]  
Note that, since \(v_{t0}^{2}-v_{tr}^{2}\) is far smaller than \(-2v_{tra}v_{s0}+v_{s0}^{2}+a_{tr}R_{0}-a_{sr}R_{0}\), it is ignored.  
From (39) one can see that the target's range displacement is caused by its radial velocity, i.e., the range displacement will be equal to zero when the radial velocity is zero. Moreover, it is interesting that the range displacement is dependent on its baseband Doppler center frequency rather than the absolute Doppler center frequency. This property indicates that the range displacement is independent of Doppler ambiguity and will not be monotonically increasing with the target's radial velocity. In addition, from (40) one can see that the range displacement is also related to the target's radial acceleration, along-track velocity, and distance. However, it is very difficult to find out the variation of the range displacement with these parameters directly from (40). To address this issue, numerical simulations are conducted, with the results being presented in Figure 4.  
Figure 4a illustrates the variation of the range displacement with the velocities of the target. It can be seen that the range displacement varies with \(v_{tr}\) periodically and gets the maximum value when the baseband Doppler center frequency equals to \(PRF/2\). In addition, one can see that the range displacement is almost independent of \(v_{ta}\). Figure 4b,c show the variation of the range displacement with the distance and radial acceleration of the target, respectively. It can be seen that the range displacement increases with \(R_{0}\) approximately linearly, and is inversely proportional to \(a_{tr}\).