## 2 Concepts and Methods
### HMS System Resolution Extraction Method Concept  
According to Figure 1, suppose the SAR platform flies in the direction of the slow-time and the point scatterer is located within the CTT with an initial slant range of \(\mathrm{R}_{0}\) from the sensor. According to Equations (18) and (24), the received 2-D backscattered pulse from the point scatterer in the case of linear frequency modulation waveform can be expressed as (40) [3]:  
\[ \begin{split}\mathrm{S}_{\mathrm{n}}(\tau_{\mathrm{r}},\mathrm{t }_{\mathrm{a}})&=\mathrm{A.~{}w}_{\tau_{\mathrm{r}}}(\tau_{ \mathrm{r}}-2\mathrm{R}(\mathrm{t}_{\mathrm{a}})/\mathrm{c}_{0})w_{\mathrm{t} _{\mathrm{a}}}(\mathrm{t}_{\mathrm{a}})\times\cos[2\pi\mathrm{k}_{0}(\tau_{ \mathrm{r}}-2\mathrm{R}(\mathrm{t}_{\mathrm{a}})/\mathrm{c}_{0})+\\ &\pi\mathrm{K}_{\mathrm{r}}(\tau_{\mathrm{r}}-2\mathrm{R}( \mathrm{t}_{\mathrm{a}})/\mathrm{c}_{0})^{2}+\psi\Big{\}}\end{split}  \tag{40}\]  
where, \(\mathrm{t}_{\mathrm{a}}\) is the slow-time, \(\tau_{\mathrm{r}}\) is the fast-time, \(\mathrm{S}_{\mathrm{n}}(\tau_{\mathrm{r}},\mathrm{t}_{\mathrm{a}})\) is the received 2-D backscattered signal from the scatterer, \(\mathrm{A}\) is the backscattering coefficient, \(w_{\tau_{\mathrm{r}}}(. )\) is the received pulse envelope in fast-time, \(w_{\mathrm{t}_{\mathrm{a}}}\) is the received pulse envelope in slow-time, \(k_{0}\) is the carrier frequency, \(\mathrm{K}_{\mathrm{r}}\) is the chirp rate, and \(\mathrm{c}_{0}\) is the speed of light. The variable of \(\psi\) is accounted for in the phase change problem during the reception procedure. According to Equations (24)-(39), the received echo is a coherent signal while being backscattered from the CTT due to the possible phase changes of \(\psi\) within the phase history [24]. The phase changes of the raw data, as mentioned in the previous sections, are considered as the source of HMS presence within the texture. The purpose of SAR system resolution extraction is twofold. The first, namely the RDG extraction, allows us to evaluate the HMS effects on the backscattering behavior of the CTT while it is being being backscattered to the sensor as \(\mathrm{S}_{\mathrm{n}}\) (\(\tau_{\mathrm{r}}\), \(\mathrm{t}_{\mathrm{a}}\)), in the form of the received backscattered signal. The second, namely the RFP extraction, carries out SAR frequency profile analysis based on the HMS presence within the \(\mathrm{S}_{\mathrm{n}}\) (\(\tau_{\mathrm{r}}\), \(\mathrm{t}_{\mathrm{a}}\)), while being arranged and reassembled into baseband contiguous radar range lines in the form of \(\psi\) interferences. The block diagram of the complete aforementioned hybrid-domain IFA based on the extended chirp scaling algorithm for the purpose of system resolution extraction is shown in Figure 6. The structure of the algorithm is prepared based on the principle of stationary phase (POSP) [3]. With the help of POSP, the corresponding baseband signal of Equation (40) is expressed as Equation (41):  
\[ \begin{split}\mathrm{S}_{\mathrm{BB}}(\tau_{\mathrm{r}},\mathrm{f }_{\mathrm{t}_{\mathrm{a}}})&=\mathrm{W}_{\tau_{\mathrm{r}}}( \tau_{\mathrm{r}}-2\mathrm{R}(\mathrm{f}_{\mathrm{t}_{\mathrm{a}}},\mathrm{r}_{ 0})/\mathrm{c}_{0}).~{}\mathrm{W}_{\mathrm{t}_{\mathrm{a}}}\big{(}\tau_{0} \mathrm{M}_{\mathrm{f}_{\mathrm{t}_{\mathrm{a}}}}/2\mathrm{v}^{2}\beta(\mathrm{ f}_{\mathrm{t}})\big{)}\\ &\times\exp[-i4\pi f_{0}\tau_{0}\beta(f_{t_{\mathrm{a}}})/\mathrm{ c}_{0}]\exp[j\pi\mathrm{K}_{\mathrm{r}}(\mathrm{f}_{\mathrm{t}_{\mathrm{a}}}, \mathrm{r}_{0})(\tau_{\mathrm{r}}-2\mathrm{R}(\mathrm{f}_{\mathrm{t}_{\mathrm{a }}},\mathrm{r}_{0})^{2}\big{]}\end{split}