## 2 Concepts and Methods
### HMS System Resolution Extraction Method Concept
 \tag{41}\]  
where \(\mathrm{v}\) is the relative velocity between the platform and the CTT, \(\lambda\) is the radar wavelength, \(\mathrm{r}_{0}\) is the distance of the closest approach, \(\mathrm{f}_{\mathrm{t}_{\mathrm{a}}}\) is the frequency corresponding to slow-time, \(\mathrm{R}(\mathrm{f}_{\mathrm{t}_{\mathrm{a}}},\mathrm{r}_{0})\) is the range migration in the time-frequency domain, and \(\mathrm{K}_{\mathrm{r}}(\mathrm{f}_{\mathrm{t}_{\mathrm{a}}},\mathrm{r}_{0})\) is the modulation rates of chirps, which are expressed by Equation (42):  
\[\begin{split}\mathrm{R}(\mathrm{f}_{\mathrm{t}_{\mathrm{a}}}, \mathrm{r}_{0})&=\frac{\mathrm{r}_{0}}{\beta(\mathrm{f}_{\mathrm{t }_{\mathrm{a}}})}=\mathrm{r}_{0}(\alpha\mathrm{a}_{\mathrm{sc}}(\mathrm{f}_{ \mathrm{t}_{\mathrm{a}}})+\alpha)\\ &\frac{1}{\mathrm{K}_{\mathrm{r}}(\mathrm{f}_{\mathrm{t}_{ \mathrm{a}}},\mathrm{r}_{0})The normal chirp scaling factor \(\alpha\) describes the Doppler frequency-dependent part of the signal and shifts as the phase centers to a new position. Therefore, a new scaling factor based on the algorithm must be applied as \(\mathrm{a_{sc}(f_{t_{a}})}\):  
\[\mathrm{a_{sc}(f_{t_{a}})}=a(f_{t_{a}})+(1-\alpha)\frac{1+a(f_{t_{a}})}{\alpha} \tag{43}\]  
The new scaling factor \(\mathrm{a_{sc}(f_{t_{a}})}\) leads to proper phase modification and is called the new phase modification factor, while \(\alpha\) is an additional scaling factor. Therefore, based on the new scaling factor \(\mathrm{a_{sc}(f_{t_{a}})}\), some phase functions like \(\mathrm{H_{1}}\), \(\mathrm{H_{2}}\), and \(\mathrm{H_{3}}\) will be applied to compensate for specific frequency variations correctly. The first compensation function is given by the following:  
\[\mathrm{H_{1}}=\exp[j\pi\mathrm{K_{r}(f_{t_{a}},r_{0})a_{sc}(f_{t_{a}})}\bigg{(} \frac{\tau_{r}-2\mathrm{R}(f_{t_{a}},r_{ref})}{c}\bigg{)}^{2}] \tag{44}\]  
where \(\mathrm{r_{ref}}\) is the reference range. With the Fourier transform in fast-time, the second phase function \(\mathrm{H2}\) will be applied directly to the signal as Equation (45):  
\[\mathrm{H_{2}}=\exp[j\pi\frac{1}{\mathrm{K_{r}(f_{t_{a}},r_{ref})(1+a_{sc}(f_{ t_{a}}))f_{\tau_{r}}}^{2}]\exp[j\frac{4\pi\tau_{ref}}{c}a(f_{t_{a}})f_{\tau_{r}}] \tag{45}\]  
where \(\mathrm{f_{\tau_{r}}}\) is the frequency corresponding to fast-time. After the inverse Fourier transform in fast-time, the compressed SAR signal is expressed by:  
Figure 6: Proposed system resolution extraction algorithm with the help of range cell migration cancellation (RCMC). \[\mathrm{S}_{\tau_{\mathrm{r}}}(\tau_{\mathrm{r}},\mathrm{f}_{\mathrm{t}_{\mathrm{a}}}) =\mathrm{A}\cdot\mathrm{W}_{\mathrm{t}}\bigg{(}-\frac{\mathrm{r}_{0} \lambda\mathrm{f}_{\mathrm{t}_{\mathrm{a}}}}{2\mathrm{v}^{2}\beta(\mathrm{f}_{ \mathrm{t}_{\mathrm{a}}})}\bigg{)}\exp[-\mathrm{i}\pi\mathrm{f}_{0}\tau_{0} \beta(\mathrm{f}_{\mathrm{t}_{\mathrm{a}}})/\mathrm{c}_{0}].\sin\mathrm{c}[ \pi\mathrm{.BW}.\tau_{\mathrm{r}}].\exp[-\mathrm{j}\Delta\varphi] \tag{46}\]  
where A is a complex constant and BW is the SAR sensor bandwidth. Therefore, the third phase function \(\mathrm{H}_{3}\) can be written as follows:  
\[\mathrm{H}_{3}=\exp[\mathrm{j}\Delta\varphi(\mathrm{f}_{\mathrm{t}_{\mathrm{a }}})] \tag{47}\]  
Before slow-time compression, another phase function is multiplied to correct the data, as follows:  
\[\mathrm{H}_{4}(\mathrm{f}_{\mathrm{t}_{\mathrm{a}}})=\exp[\mathrm{j}\frac{4\pi }{\lambda}\bigg{(}\mathrm{r}_{\mathrm{ref}}+\frac{(\mathrm{r}_{0}-\mathrm{r} _{\mathrm{ref}})}{\alpha}\bigg{)}(\beta(\mathrm{f}_{\mathrm{t}_{\mathrm{a}}}) -1)\bigg{]} \tag{48}\]  
Therefore, according to the equations derived from Equation (41) to Equation (48), on the basis of Figure 6 and the geometry mentioned in Figure 1, the inversed procedure of the proposed algorithm will be applied step by step to the SAR de-speckled images for the purpose of system resolution extractions and comparison.