# A Novel Spaceborne Sliding Spotlight Range Sweep Synthetic Aperture Radar: System and Imaging
## 4 System Parameter Design
### Carrier Frequency and Chirp Rate  
By comparing Figure 5a,b, it is clear that due to the serious spatial variation of the range wavenumber, a large ratio of data, marked by the grey regions, have been wasted, resulting in a much compressed range bandwidth of \(K_{\text{Y}}\) and a degraded range resolution. In an extreme case where \(K_{\text{Ymax}}\) is smaller than \(K_{\text{Ymin}}\), it is even impossible to implement focusing.  
Based on Equation (21), the spatial variation of \(K_{\rm Y}\) is caused by the variations of the instant look angle \(\beta\) and azimuth angle \(\alpha\) and the azimuth-dependent time delay \(\tau_{\rm d}\). These two kinds of variations have different impacts on \(K_{\rm Y}\). As the angle-related range wavenumber variation is range-independent, it can be fully counteracted by using the PA method discussed in [(23)]. On the other aspect, the range wavenumber variation caused by \(\tau_{\rm d}\) relies on both the range and azimuth, hence cannot be fully compensated by the PA method. As a result, the echo cannot be directly received with the Keystone format distribution, as shown in Figure 5b, by adjusting the parameters pulse by pulse. However, we can develop a new PA technique based on the expression of the range wavenumber to suppress the spatial variation of \(K_{\rm Y}\) and to minimize the loss of the range wavenumber bandwidth while conducting the range interpolation.  
Similar to the PA technique in [(23)], the new PA technique adjusts the carrier frequency \(f_{\rm c}\) and the chirp rate \(\gamma\) pulse by pulse based on the instant data acquisition geometry as  
\[\left\{\begin{array}{l}f_{\rm c}(m)=f_{\rm 0}\frac{\sin\beta_{\rm c}}{\sin \beta(m)\cos a(m)}\cdot\frac{f_{\rm 0}+\gamma_{\rm 0}\Big{(}-\frac{2R_{\rm pdf}(N_{ \rm a}/2)}{c}+\tau_{\rm d}(N_{\rm a}/2)\Big{)}}{f_{\rm 0}+\gamma_{\rm 0} \Big{(}-\frac{2R_{\rm pdf}(m)}{c}+\tau_{\rm d}(m)\Big{)}}\\ \\ \gamma(m)=\gamma_{\rm 0}\frac{\sin\beta_{\rm c}}{\sin\beta(m)\cos a(m)} \cdot\frac{f_{\rm 0}+\gamma_{\rm 0}\Big{(}-\frac{2R_{\rm pdf}(N_{\rm a}/2)}{c}+\tau_{ \rm d}(N_{\rm a}/2)\Big{)}}{f_{\rm 0}+\gamma_{\rm 0}\Big{(}-\frac{2R_{\rm pdf}(m)}{c}+ \tau_{\rm d}(m)\Big{)}}\end{array}\right. \tag{46}\]  
where \(f_{\rm c0}\) and \(\gamma_{\rm 0}\) denote the constant referential carrier frequency and the constant referential chirp rate, respectively. The first action is to counteract the angle-related spatial variation and the second action is to suppress the spatial variation caused by \(\tau_{\rm d}\).  
By applying the new PA technique in Equation (46), the recorded polar format data are shown in Figure 6a, where the spatial variation of the range wavenumber has been greatly suppressed. In this case, the Keystone format data generated by the range interpolation, as shown in Figure 6b, suffer from a much less range wavenumber bandwidth loss. The rectangular format data after the azimuth interpolation are shown in Figure 6c. Comparing Figures 5 and 6, it is clear that the ratio of the wasted data has been effectively decreased benefiting from the new PA manipulation in Equation (46).