# A Novel Spaceborne Sliding Spotlight Range Sweep Synthetic Aperture Radar: System and Imaging
## 3 Modified Polar Format Algorithm
### Modified Two Dimensional Dechirp  
While the SSS-RSSAR does not apply the dechirp-on-receive mechanism, the chirp characteristic of the echo should be removed firstly, along both the range and azimuth dimensions. In the case of a transmitted LFM signal, the backscattered echo of an arbitrary Target \(S\) can be digitally expressed as  
Figure 4: Echoes before and after the zero padding if processed by the algorithms such as the Range Doppler Algorithm (RDA), Chirp Scaling Algorithm (CSA) and Range Migration Algorithm (RMA).  
\[\begin{split}& E_{\text{s}}(m,i)=\text{rect}\Big{[}\frac{m-m_{\text{s}}}{N _{\text{as}}}\Big{]}\text{rect}\Big{[}\frac{i}{T_{\text{p}}F_{\text{s}}}-\frac{2 R_{\text{s}}(m)}{cT_{\text{p}}}+\frac{\tau_{\text{d}}(m)}{T_{\text{p}}}\Big{]}\\ &\cdot\exp\biggl{\{}j\pi\gamma\Big{(}\frac{i}{F_{\text{s}}}-\frac{2 R_{\text{s}}(m)}{c}+\tau_{\text{d}}(m)\Big{)}^{2}\biggr{\}}\exp\Bigl{\{}-j\frac{4\pi f _{\text{c}}R_{\text{s}}(m)}{c}\Bigr{\}},\\ & m=-\frac{N_{\text{a}}}{2},...,\frac{N_{\text{a}}}{2},i=-\frac{ N_{\text{r}}}{2},...,\frac{N_{\text{r}}}{2}\end{split} \tag{9}\]  
where \(m\) and \(i\) denote the azimuth and range sample indexes, respectively; \(f_{\text{c}}\) denotes the carrier frequency; \(\gamma\) denotes the chirp rate of the LFM signal; \(T_{\text{p}}\) denotes the pulse width; \(F_{\text{s}}\) denotes the sampling rate; and \(R_{\text{s}}\) denotes the instant slant range for Target \(S\). \(N_{\text{a}}\) and \(N_{\text{r}}\) denote the azimuth and the range sample numbers, respectively. \(N_{\text{as}}\) denotes a smaller-than-\(N_{\text{a}}\) azimuth sample number for Target \(S\). \(m_{\text{s}}\) denote azimuth position where the beam center illuminates Target \(S\). \(\tau_{\text{d}}\) denotes the time delay. While \(\tau_{\text{d}}\) is a constant for the traditional constant-PI SAR, for the SSS-RSSSAR, \(\tau_{\text{d}}\) is azimuth-sensitive due to the CVPI technique as  
\[\tau_{\text{d}}(m)=\sum_{k=m-M}^{m-1}PI(k)+\frac{T_{\text{p}}}{2} \tag{10}\]  
Based on the echo in Equation (9), the new 2D dechirp filter of the MPFA yields  
\[F_{\text{dechirp}}(m,i)=\exp\biggl{\{}-j\pi\gamma\biggl{(}\frac{i}{F_{\text{s }}}-\frac{2R_{\text{ref}}(m)}{c}+\tau_{\text{d}}(m)\biggr{)}^{2}\biggr{\}}\exp \biggl{\{}j\frac{4\pi f_{\text{c}}R_{\text{ref}}(m)}{c}\biggr{\}} \tag{11}\]  
where \(R_{\text{ref}}\) denotes the instant slant range from the antenna to the swath center. By multiplying the echo in Equation (9) with \(F_{\text{dechirp}}\), the dechirped signal yields  
\[\begin{split}& E_{\text{s1}}(m,i)=\text{rect}\Big{[}\frac{m-m_{ \text{s}}}{N_{\text{as}}}\Big{]}\text{rect}\Big{[}\frac{i}{T_{\text{p}}F_{ \text{s}}}-\frac{2R_{\text{s}}(m)}{cT_{\text{p}}}+\frac{\tau_{\text{d}}(m)}{T_ {\text{p}}}\Big{]}\\ &\cdot\exp\Bigl{\{}-j\frac{4\pi}{c}\Bigl{(}f_{\text{c}}+\gamma \Bigl{(}\frac{i}{F_{\text{s}}}-\frac{2R_{\text{ref}}(m)}{c}+\tau_{\text{d}}(m) \Bigr{)}\Bigr{)}\Delta R_{\text{s}}(m)\Bigr{\}}\exp\Bigl{\{}j\frac{4\pi\gamma \lambda R_{\text{s}}^{2}(m)}{c^{2}}\Bigr{\}}\end{split} \tag{12}\]  
where \(\Delta R_{\text{s}}\) denotes differential slant range as  
\[\Delta R_{\text{s}}(m)=R_{\text{s}}(m)-R_{\text{ref}}(m) \tag{13}\]