# An Efficient Imaging Method for Medium-Earth-Orbit Multichannel SAR-GMTI Systems
## 4 Property Analysis of Moving Target in the Complex Image Domain
### Along-Track Interferometry Phase  
ATI phase is a very important parameter in the SAR-GMTI field. It can be utilized to estimate the radial velocity of the target. In addition, moving targets can be detected according to the interference phase diagram. Based on (37), the signal of the second channel after registration is:  
\[\begin{array}{rl}S_{2,{\alpha},{\alpha},{\alpha}}(t_{r},f_{a})&={W_{a}}(f_{a })p_{r}\Bigg{\{} \begin{array}{l}t_{r}-\frac{2}{6}\left(R_{0}-\frac{(1+4a)^{2 }}{8}-\frac{(1+4a)^{3}}{82}\right)-\frac{(1+4a)^{3}}{82}\frac{1}{2}\left(2f_{a }\cdot M\cdot PRF+M^{2}\cdot PRF^{2}\right)\\ -\frac{c}{f_{a}^{2}}\frac{3(l_{1}+4a)f_{3}}{162}\left(2f_{a}\cdot M\cdot PRF+M ^{2}\cdot PRF^{2}\right)-\frac{d^{2}}{f_{a}^{2}}\frac{1}{162}\left(M^{3}\cdot PRF ^{3}+3f_{a}^{2}M\cdot PRF+3f_{a}\cdot M^{2}\cdot PRF^{2}\right)\\ \times\exp\left\{j\pi\left(-\frac{4R_{0}}{2}+\frac{(1+4a)^{2}}{24}\frac{(1+4a) ^{2}}{24}\right)\right\}\exp\left\{j\pi\left(\frac{4+4a}{2}+\frac{3(l_{1}+4a)^ {2}}{4}\right)f_{a}\right\}\\ \times\exp\left\{j\pi\cdot M\cdot PRF\cdot\left(\frac{1+4a}{2}+\frac{3(l_{1}+4a) ^{2}f_{3}}{42}+\frac{\lambda}{42}\cdot M\cdot PRF+\lambda\frac{3(l_{1}+4a)f_{ 3}}{82}\cdot M\cdot PRF+\frac{1\lambda^{2}}{162}\cdot M^{2}\cdot PRF^{2}\right) \right\}\\ \times\exp\left\{j\pi\cdot M\cdot PRF\left(\frac{\lambda_{1}}{2}+\frac{3(l_{1}+ 4a)f_{3}}{42}+\frac{3(l_{2}+4a)}{162}\cdot M\cdot PRF\right)f_{a}\right\}\\ \times\exp\left\{j\pi\lambda\left[\left(\frac{1}{2}-\frac{1}{24}\right)f_{a}^{ 2}+\frac{3(l_{1}+4a)f_{3}}{82}-\frac{3(l_{2}+4a)}{82}\right)f_{a}^{2}+\left( \frac{\lambda_{1}}{162}-\frac{\lambda_{1}}{162}\right)f_{a}^{3}\right]\right\} \exp\left\{j\pi\frac{3l_{1}\lambda^{2}}{162}\cdot M\cdot PRF\cdot f_{a}^{2} \right\}.\end{array}  \tag{43}\]  
According to (37) and (43), the interference phase between channel 1 and channel 2 can be expressed as  
\[ \begin{array}{rl}\phi_{ATI_{12}}&=\arg\big{(}S_{1,ac}(t_{r},f_{a})S_{2,ac, reg}{}^{*}(t_{r},f_{a})\big{)}\\ &=-\frac{2\pi l_{1}da}{\lambda_{2}}-\frac{\pi d^{2}a^{2}}{\lambda_{2}}-\frac{3 \pi l_{1}^{2}da\lambda_{1}}{2\lambda_{2}^{2}}-\frac{3\pi l_{1}l_{3}d^{2}a^{2}}{2 \lambda_{2}^{2}}-\frac{\pi d^{2}a^{3}l_{1}}{2\lambda_{2}^{2}}-\frac{\pi d^{2} a^{3}l_{2}}{2\lambda_{2}^{2}}-\frac{\pi d\alpha}{l_{2}}\left(f_{a}+M\cdot PRF \right)\\ &-\frac{6\pi l_{1}da}{42}\left(f_{a}+M\cdot PRF\right)-\frac{3\pi d^{2}a^{2}l_{ 3}}{42}\left(f_{a}+M\cdot PRF\right)\\ &-\pi\lambda\frac{3d\alpha l_{1}}{82}\left(f_{a}+M\cdot PRF\right)^{2}-2\pi\frac {d}{v_{\alpha 0}}\left(f_{a}+M\cdot PRF\right),\end{array}  \tag{44}\]  
where \({}^{*}\) represents a conjugate operation on the signal. Since the items containing \(d^{2}a^{2}\), \(d^{3}a^{3}\) and \(l_{3}\) are much smaller than other items, they can be ignored. Therefore, \(\phi_{ATI_{12}}\) can be simplified to  
\[\phi_{ATI_{12}}\approx-\frac{\pi d\alpha}{l_{2}}M\cdot PRF+2\pi\frac{d}{v_{s0} }\frac{2}{\lambda}v_{tr}-2\pi\frac{d}{v_{s0}}M\cdot PRF. \tag{45}\]  
In order to find out the variation of ATI phase with the target's parameters, the expressions of \(\alpha\) and \(l_{2}\) are substituted into (45). Then, the ATI phase can be rewritten as:  
\[\phi_{ATI_{12}}=\frac{4\pi d\upsilon_{tr}}{\lambda v_{s0}}-\frac{2\pi d}{v_{s0} }M\cdot PRF-\pi d\frac{2(v_{ta}-v_{s0})v_{s0}+2a_{sr}R_{0}}{(-2v_{ta}v_{s0}+v_{s 0}^{2}+a_{tr}R_{0}-a_{sr}R_{0})v_{s0}}M\cdot PRF. \tag{46}\]  
It can be seen from (46) that the ATI phase is also caused by the target's radial velocity and is only dependent on the radial velocity when there is no Doppler ambiguity. For the cases where Doppler ambiguity is happening, in addition to the radial velocity the ATI phase depends also on the along-track velocity, radial acceleration and distance. To find out the variation of the ATI phase with these parameters, numerical experiments are carried out, with the results being presented in Figure 6. Figure 6a shows the relationship between the ATI phase and velocities.