# A Novel Spaceborne Sliding Spotlight Range Sweep Synthetic Aperture Radar: System and Imaging
## 3 Modified Polar Format Algorithm
### Interpolation-Based Data Format Correction  
As the 2D dechirp has uniquely mapped the signal from the time domain to the frequency domain, the range-azimuth signal coupling can be removed by the interpolation. Firstly, a new line-of-sight wavenumber \(K_{\text{R}}\) is defined as  
\[K_{\text{R}}(i)=\frac{4\pi}{c}\bigg{(}f_{\text{c}}+\gamma\bigg{(}\frac{i}{F_{ \text{s}}}-\frac{2R_{\text{ref}}(m)}{c}+\tau_{\text{d}}(m)\bigg{)}\bigg{)} \tag{19}\]  
Different from the conventional wavenumber, the new wavenumber in Equation (19) contains the time delay induced by the CVPI technique. By substituting Equation (19) into Equation (18) and neglecting the amplitude terms, Equation (18) can be rewritten as  
\[E_{\text{s2}}(m,i)=\exp\{-jK_{\text{R}}(i)\Delta R_{\text{s}}(m)\} \tag{20}\]  
The projections of \(K_{\text{R}}\) along the X- and Y-axis, denoted as \(K_{\text{X}}\) and \(K_{\text{Y}}\), respectively, yield  
\[\left\{\begin{array}{l}K_{\text{X}}(m,i)=K_{\text{R}}(i)\sin\beta(m)\sin \alpha(m)\\ K_{\text{Y}}(m,i)=K_{\text{R}}(i)\sin\beta(m)\cos\alpha(m)\end{array}\right. \tag{21}\]  
By assuming that the transmitted signal has a planar waveform, the differential slant range \(\Delta R_{\text{s}}\) can be approximated as [3; 23].  
\[\Delta R_{\text{s}}(m)\approx-\sin\beta(m)\sin\alpha(m)x_{\text{s}}-\sin\beta (m)\cos\alpha(m)y_{\text{s}} \tag{22}\]  
where \(x_{\text{s}}\) and \(y_{\text{s}}\) are the X and Y coordinates of Target \(S\). By substituting Equation (22) into Equation (20), \(E_{\text{s2}}\) can be approximated as  
\[E_{\text{s2}}(m,i)\approx\exp\{j(K_{\text{X}}(m,i)x_{\text{s}}+K_{\text{Y}}(m, i)y_{\text{s}})\} \tag{23}\]  
By assuming three targets, denoted as Target \(S_{1}\), \(S_{2}\) and \(S_{3}\), are located along the X-axis in sequence, their echoes after the RVP removal processing will be distributed in a polar format in the wavenumber as shown in Figure 5a. Then a range interpolation is applied to eliminate the spatial variation of the range wavenumber as  
\[K_{\text{Y}}(m,i)\to K^{\prime}_{\text{Y}}(i) \tag{24}\]where \(K^{\prime}{}_{Y}\) denotes the referential range wavenumber as  
\[K^{\prime}_{Y}(i)=K_{Ymin}+\frac{K_{Ymax}-K_{Ymin}}{N_{r}-1}\bigg{(}i+\frac{N_{r }}{2}\bigg{)} \tag{25}\]  
where \(K_{Ymin}\) and \(K_{Ymax}\) denote the minimum and maximum values of \(K^{\prime}{}_{Y}\), respectively, as  
\[\left\{\begin{array}{l}K_{Ymin}=\max\limits_{m=-\frac{N_{\rm a}}{2},\ldots, \frac{N_{\rm a}}{2}}(K_{Y}(m,-N_{r}/2))\\ K_{Ymax}=\min\limits_{m=-\frac{N_{\rm a}}{2},\ldots,\frac{N_{\rm a}}{2}}(K_{Y} (m,N_{r}/2))\end{array}\right. \tag{26}\]  
After the range interpolation, the signal is converted from the polar format to the keystone format as shown in Figure 5b, as  
\[E_{\rm s3}(m,i)=\exp\big{\{}j\big{(}K^{\prime}_{Y}(i)\tan\alpha(m)x_{\rm s}+K^ {\prime}_{Y}(i)y_{\rm s}\big{)}\big{\}} \tag{27}\]  
Then an azimuth interpolation is employed to convert the data from the Keystone format to the rectangular format as  
\[K^{\prime}_{Y}(i)\tan\alpha(m)\to K^{\prime}_{X}(m) \tag{28}\]  
where the azimuth referential wavenumber \(K^{\prime}{}_{X}\) yields  
\[K^{\prime}_{X}(m)=K_{Xmin}+\frac{K_{Xmax}-K_{Xmin}}{N_{\rm a}-1}\bigg{(}m+\frac {N_{\rm a}}{2}\bigg{)} \tag{29}\]  
where \(K_{Xmin}\) and \(K_{Xmax}\) denote the minimum and maximum values of \(K^{\prime}{}_{X}\), respectively, as  
\[\left\{\begin{array}{l}K_{Xmin}=K_{Ymin}\tan\alpha\bigg{(}-\frac{N_{\rm a}}{ 2}\bigg{)}\\ K_{Xmax}=K_{Ymin}\tan\alpha\bigg{(}\frac{N_{\rm a}}{2}\bigg{)}\end{array}\right. \tag{30}\]  
The resulted rectangular-format signal is shown in Figure 5c and can be expressed as  
\[E_{\rm s4}(m,i)=\exp\big{\{}j\big{(}K^{\prime}_{X}(m)x_{\rm s}+K^{\prime}_{Y} (i)y_{\rm s}\big{)}\big{\}} \tag{31}\]  
By conducting the FFT along both the range and azimuth dimensions, the final focused image can be achieved.  
Figure 5: Wavenumber data distributed in: (**a**) polar format; (**b**) keystone format; and (**c**) rectangular format.