## 3 The Proposed Method
### Focused Imaging and Relocation  
After the motion parameter estimation is completed, the focused imaging of moving objects can be achieved through two steps of RCMC and matched filtering in the azimuth direction. For Equation (6), the range migration phase compensation factor constructed from the estimates of \(a_{1}\), \(a_{2}\) and \(a_{3}\) can be written as [10]  
\[H_{rcmc}(f_{r},t_{a})=Aw_{a}(t_{a})rect\left(\frac{f_{r}}{B}\right)\exp\left\{ j\frac{4\pi f_{r}}{c}\left(a_{1}t_{a}+a_{3}t_{a}^{2}+a_{3}t_{a}^{3}\right)\right\} \tag{32}\]  
By combining Equations (6) and (32), the range-frequency domain and azimuth-time domain signal after the RCMC can be written as  
\[\begin{split} S_{7}(f_{r},t_{a})&=S_{1}(f_{r},t_{ a})H_{rcmc}(f_{r},t_{a})\\ &=Aw_{a}(t_{a})rect\left(\frac{f_{r}}{B}\right)\exp\left\{-j \frac{4\pi f_{r}R_{0}}{c}\right\}\\ &\times\exp\left\{-j\frac{4\pi f_{c}}{c}\left(R_{0}+a_{1}t_{a}+a _{2}t_{a}^{2}+a_{3}t_{a}^{3}\right)\right\}\end{split} \tag{33}\]  
Performing IFFT in the range direction on Equation (33), the two-dimensional time domain signal after RCMC can be obtained as  
\[\begin{split} S_{7}(t_{r},t_{a})&=Aw_{a}(t_{a}) sinc\left[B\left(t_{r}-\frac{2R_{0}}{c}\right)\right]\\ &\times\exp\left\{-j\frac{4\pi f_{c}}{c}\left(R_{0}+a_{1}t_{a}+a _{2}t_{a}^{2}+a_{3}t_{a}^{3}\right)\right\}\end{split} \tag{34}\]  
According to the signal obtained by Equation (34), an azimuth matched filter is constructed to perform azimuth focusing and positioning. The azimuth matched filter can be obtained with Equation (35).  
\[H_{match}(t_{r},f_{a})=FFT_{t_{a}}\left\{w_{a}(t_{a})\exp\left\{-j\frac{4\pi f _{c}}{c}\left(a_{1}t_{a}+a_{2}t_{a}^{2}+a_{3}t_{a}^{3}\right)\right\}\right\} ^{*} \tag{35}\]  
where \"\(\bullet\)\" is the conjugate.  
After matching the defocusing effects caused by the RCMC and performing the azimuth FFT, we finally obtain the imaging result in the two-dimensional time domain as follows  
\[\begin{split} S_{image}(t_{r},t_{a})&=IFFT_{f_{a}} \left\{S_{7}(t_{r},f_{a})\cdot H_{match}(t_{r},f_{a})\right\}\\ &=Asinc\left[B\left(t_{r}-\frac{2R_{0}}{c}\right)\right]sinc \left[\frac{4a_{2}T_{s}}{\lambda}t_{a}\right]\end{split} \tag{36}\]  
In the imaging result of the stationary scene without processing of GMTI, targets moving along the radial direction appear displaced in azimuth from their original position. The targets moving with radial velocity are shifted in azimuth by [1]  
\[\Delta_{az}=-R\frac{v_{r}}{|v|} \tag{37}\]where \(R\) is the distance from the sensor to the targets.  
In the above analysis, the proposed method is derived in the scenario with a single target. However, it can also be extended to the scenario with multiple targets. Firstly, moving-target focusing imaging is carried out after clutter suppression and target detection. Therefore, we can separate the multiple targets according to their different moving trajectories. Moreover, if the scattering intensities of different targets differ significantly, the 'CLEAN' technique could be applied to eliminate the effect of the strong target [14; 21]. In this way, the coherent integration of strong moving targets and weak ones can be achieved iteratively.