# A Novel Spaceborne Sliding Spotlight Range Sweep Synthetic Aperture Radar: System and Imaging
## 3 Modified Polar Format Algorithm
### Residual Video Phase Removal  
The second phase term in Equation (12) denotes the Residual Video Phase (RVP) that impacts SAR imagery because the displacement between the two waveforms varies with time as the two-way range to a scatterer changes over a coherent processing interval [3]. In the time domain the RVP is hard to compensate because \(\Delta R_{\text{s}}\) is space-variant. However, the RVP can be conveniently compensated in the frequency domain.  
By making a range Fast Fourier Transform (FFT) along the \(i\) dimension, the range spectrum of the signal in Equation (12) yields  
\[\begin{split}& E_{\text{s1}}(m,i^{\prime})=\text{rect}\Big{[} \frac{m-m_{\text{s}}}{N_{\text{as}}}\Big{]}\sin c\Big{[}\frac{T_{\text{p}}F_{ \text{s}}}{N_{\text{r}}}\Big{(}i^{\prime}+\frac{2\gamma\Delta R_{\text{s}}(m)N_{ \text{r}}}{cF_{\text{s}}}\Big{)}\Big{]}\\ &\cdot\exp\Bigl{\{}-j\frac{4\pi\gamma\Delta R_{\text{s}}^{2}(m)}{c^ {2}}\Bigr{\}}\exp\Bigl{\{}-j\frac{4\pi F_{\text{s}}\Delta R_{\text{s}}(m)}{cN_ {\text{r}}}i^{\prime}\Bigr{\}}\\ &\cdot\exp\Bigl{\{}-j\frac{4\pi\gamma}{c}\Bigl{(}f_{\text{c}}+ \gamma\Bigl{(}-\frac{2R_{\text{ref}}(m)}{c}+\tau_{\text{d}}(m)\Bigr{)}\Bigr{)} \Delta R_{\text{s}}(m)\Bigr{\}}\exp\Bigl{\{}-j\frac{2\pi F_{\text{s}}}{N_{\text {r}}}\Bigl{(}\frac{2R_{\text{ref}}(m)}{c}-\tau_{\text{d}}(m)\Bigr{)}i^{\prime }\Bigr{\}}\end{split} \tag{14}\]  
Based on the \(\text{sinc}(\cdot)\) function in Equation (14), the target is found to be focused at \(i^{\prime}\)s as  
\[i^{\prime}_{\text{s}}=-\frac{2\gamma N_{\text{r}}\Delta R_{\text{s}}}{cF_{\text {s}}} \tag{15}\]By substituting Equation (15) into Equation (14), the latter equation can be rewritten as  
\[\begin{split}& E_{\text{s1}}(m,i^{\prime})=\text{rect}\Big{[} \frac{m-m_{\text{s}}}{N_{\text{ss}}}\Big{]}\sin c\Big{[}\frac{T_{\text{P}}F_{ \text{s}}}{N_{\text{r}}}(i^{\prime}-i^{\prime}_{\text{s}})\Big{]}\exp\Big{\{} j\frac{2\pi f_{\text{c}}F_{\text{s}}i^{\prime}_{\text{s}}}{\gamma N_{\text{r}}} \Big{\}}\\ &\cdot\exp\Big{\{}j\frac{\pi F_{\text{s}}^{2}i^{\prime 2}}{\gamma N_{ \text{r}}^{2}}\Big{\}}\exp\Big{\{}j2\pi\Big{(}-\frac{2R_{\text{ref}}(m)}{c}+ \tau_{\text{d}}(m)\Big{)}\frac{F_{\text{s}}(i^{\prime}_{\text{s}}+i^{\prime} )}{N_{\text{r}}}\Big{\}}\end{split} \tag{16}\]  
Aiming at compensating the second quadratic phases in Equation (16) at \(i^{\prime}=i^{\prime}_{\text{s}}\), the RVP removal filter can be generated as  
\[F_{\text{RVP}}\big{(}m,i^{\prime}\big{)}=\exp\left\{-j\frac{\pi F_{\text{s}}^{ 2}i^{\prime 2}}{\gamma N_{\text{r}}^{2}}\right\} \tag{17}\]  
By multiplying \(F_{\text{RVP}}\) with range spectrum of \(E_{\text{s1}}\) and converting the signal back to the range time domain via the range Inverse FFT (IFFT), the signal yields  
\[\begin{split} E_{\text{s2}}(m,i)&=\text{rect}\Big{[} \frac{m-m_{\text{s}}}{N_{\text{ss}}}\Big{]}\text{rect}\Big{[}\frac{i}{T_{\text {P}}F_{\text{s}}}-\frac{2R_{\text{ref}}(m)}{c\,\Gamma_{\text{P}}}+\frac{\tau_ {\text{d}}(m)}{\Gamma_{\text{P}}}\Big{]}\\ &\cdot\exp\Big{\{}-j\frac{4\pi}{c}\Big{(}f_{\text{c}}+\gamma \Big{(}\frac{i}{F_{\text{s}}}-\frac{2R_{\text{ref}}(m)}{c}+\tau_{\text{d}}(m) \Big{)}\Big{)}\Delta R_{\text{s}}(m)\Big{\}}\end{split} \tag{18}\]