## 4 HRWS Imaging Processing
### SAR Range Model  
The second-order and fourth-order Taylor series expansion are executed to approximate the instantaneous range of Equation (2) respectively, then the hyperbolic range model and higher-order range model are given as  
\[R_{a1}(\eta)=R_{0}+\frac{Lr\omega_{s}^{2}\eta^{2}}{2R_{0}}, \tag{16}\]  
\[R_{a2}(\eta)=R_{0}+\frac{Lr\omega_{s}^{2}\eta^{2}}{2R_{0}}-\frac{Lr\omega_{s} ^{4}\eta^{4}}{24R_{0}}-\frac{L^{2}r^{2}\omega_{s}^{4}\eta^{4}}{8R_{0}^{3}}, \tag{17}\]  
where \(R_{0}=\sqrt{\left(r-L\right)^{2}+h^{2}}\). According to the SAR system parameters setting of the hypersonic platform, as shown in Table 1, the azimuth phase errors resulted from the approximate expansion of the slant range are calculated, respectively. As shown in Figure 4a, the expression of the hyperbolic range model is relatively simple, thus it is helpful to execute the subsequent imaging processing and improve the calculation efficiency; however, the introduced azimuth phase error is more than \(\pi/4\), which results in the degradation of the imaging quality. Moreover, when the fourth-order series expansion is used to approximate the instantaneous slant range, as shown in Figure 4b, the introduced azimuth phase error is less than \(\pi/4\), thus the effects on subsequent imaging processing can be ignored. However, the high-order form of the SAR slant range is relatively complex, which results in high computational complexity for SAR imaging. Therefore, the optimal approximate processing of the cosine function is carried out further. Based on the SAR imaging geometry model shown in Figure 1, the azimuth dimension can be redefined as an angular variation varying with the azimuth time, that is, \(\theta(\eta)=\omega_{s}\eta\). Then, Equation (2) is rewritten as  
\[R_{a}(\theta)=\sqrt{L^{2}+r^{2}-2 Lr\cos\theta+h^{2}}. \tag{18}\]  
The cosine function in Equation (18) is further constructed as a second-order polynomial form, which is  
\[\cos\theta=\beta_{0}+\beta_{1}\theta^{2}. \tag{19}\]  
Based on Equation (19), the solving of fitting coefficients \(\beta_{0}\) and \(\beta_{1}\) will be transformed into a minimax optimization problem, and its optimal solution is the minimum over feasible domain \(\beta\) of the maximum over feasible domain \(\theta\) on the objective function \(\Gamma(\beta,\theta)\). Mathematically, this minimax optimization problem is formally defined as  
\[\min_{\beta_{0},\beta_{1}}\max_{\theta}\Gamma(\beta,\theta)=\min_{\beta_{0}, \beta_{1}}\max_{\theta}\Bigl{\lvert}\cos\theta-\Bigl{(}\beta_{0}+\beta_{1} \theta^{2}\Bigr{)}\Bigr{\rvert}, \tag{20}\]  
where the search interval of \(\theta\) is set as \(Y_{\theta}=[-\alpha_{theta}/2,\alpha_{theta}/2]\); here, \(\alpha_{theta}\) is the azimuth beam width of the single channel SAR system. Moreover, the search interval of \(\beta_{0}\) and \(\beta_{1}\) are set as \(X_{\beta}=[-5,5]\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Platform velocity & 6 Ma \\ Platform height & 60 km \\ Turning radius & 100 km \\ Center of slant range & 131 km \\ Frequency & 10 GHz \\ Signal bandwidth & 150 MHz \\ Pulse width & 10 us \\ Oversampling factor (azimuth) & 1.2 \\ Oversampling factor (range) & 1.4 \\ Beamwidth (sub-aperture) & 2.8648\({}^{\circ}\) \\ Length of sub-aperture & 0.6 m \\ Number of sub-apertures & 7 \\ Doppler Bandwidth & 6800 Hz \\ Ideal PRF & 1165.7 Hz \\ System PRF & 1398.9 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters setting of SAR system. Figure 4: SAR azimuth phase errors introduced by approximate expansion of slant range: (**a**) Second-order Taylor series expansion; (**b**) fourth-order Taylor series expansion.