## 4 HRWS Imaging Processing
### ETF Imaging Algorithm  
Based on the derived optimal slant range model of the hypersonic platform SAR, the ETF imaging algorithm is further designed to process the echo signal of the equivalent single channel SAR system. Assume that DPCAM SAR for hypersonic platform transmits the chirp signal. Consequently, the two-dimensional baseband signal can be given as  
\[ \begin{array}{ll}s_{arc}(\tau,\eta)&=W_{1}\exp\left(j\pi K_{r}(\tau-2R_{a}( \eta)/c)^{2}\right)\widetilde{h}_{a}(\eta)\\ &=W_{1}\exp\left(j\pi K_{r}(\tau-2R_{a}(\eta)/c)^{2}\right)\exp(-j\pi 4\pi R_{a}( \eta)/\lambda)\end{array} , \tag{21}\]  
where \(W_{1}=\sigma_{r}w_{r}(\tau-2R_{a}(\eta)/c)w_{a}(\eta-\eta_{c})\), and \(\sigma_{r}\) is the backscatter coefficient of the targets, \(\lambda\) is the wavelength, \(c\) denotes the speed of light, \(K_{r}\) represents the range frequency rate, \(\tau\) accounts for the range time, and \(w_{r}\) and \(w_{a}\) are the envelope of the range and azimuth, respectively. The range Fourier transform of Equation (21) is performed using principle of stationary phase (POSP); then, the two-dimensional signal in range frequency domain and azimuth time domain is given by  
\[S_{arc}(f_{\tau},\eta)=W_{2}\exp(-j4\pi(f_{c}+f_{\tau})R_{a}(\eta)/c)\exp\left( -j\pi f_{\tau}^{2}/K_{r}\right), \tag{22}\]  
where \(W_{2}=\sigma_{r}w_{r}(f_{\tau}/K_{r})w_{a}(\eta-\eta_{c})\), \(f_{\tau}\) and \(f_{c}\) represent the range frequency and carrier frequency, respectively. Furthermore, a closed form solution of the azimuth Fourier transform of Equation (22) is derived using POSP again. Here, the azimuth Fourier transform is given by  
\[ \begin{array}{ll}S_{arc}(f_{\tau},f_{\eta})&=\int_{-\infty}^{\infty}S_{arc }(f_{\tau},\eta)\exp\left(-j2\pi f_{\eta}\eta\right)d\eta\\ &=\int_{-\infty}^{\infty}\exp\left(-j\pi\left(4(f_{c}+f_{\tau})R_{a}(\theta)/ c+2f_{\eta}\theta/\omega_{s}+f_{\tau}^{2}/K_{r}\right)\right)d\theta\end{array} , \tag{23}\]  
where \(f_{\eta}\) is the azimuth frequency. From Equation (23), the phase of the Fourier integrand is  
\[\Theta=-\frac{4\pi(f_{c}+f_{\tau})R_{a}(\theta)}{c}-2\pi f_{\eta}\frac{\theta }{\omega_{s}}-\frac{\pi f_{\tau}^{2}}{K_{r}}. \tag{24}\]  
Figure 6: The phase errors introduced by the approximate polynomial fitting of the SAR slant range. To further simplify the calculation, the range variable is set as \(R_{s}=\sqrt{L^{2}+r^{2}-2 Lr\beta_{0}+h^{2}}\); then, Equation (18) is rewritten as  
\[R_{a}(\theta)=\sqrt{R_{s}^{2}-2 Lr\beta_{1}\theta^{2}}. \tag{25}\]  
Substituting Equation (25) into Equation (24), we can obtain the derivative of \(\Theta\) with respect to \(\theta\), i.e.,  
\[\frac{d\Theta}{d\theta}=\frac{8\pi(f_{c}+f_{\tau})Lr\beta_{1}\theta}{c\sqrt{R _{s}^{2}-2 Lr\beta_{1}\theta^{2}}}-2\pi\frac{f_{\eta}}{\omega_{s}}. \tag{26}\]  
When \(d\Theta/d\theta=0\), the stationary phase angle is  
\[\theta^{*}=\frac{cf_{\eta}R_{s}}{4(f_{c}+f_{\tau})Lr\beta_{1}\omega_{s}\sqrt{ 1+c^{2}f_{\eta}^{2}/\left(8(f_{c}+f_{\tau})^{2}Lr\beta_{1}\omega_{s}^{2} \right)}}. \tag{27}\]  
Substituting Equation (27) into Equation (23), we can get the approximate two-dimensional spectrum of DPCAM SAR for hypersonic platform as follows:  
\[S_{arc}(f_{\tau},f_{\eta})=W_{3}\exp\left(-j\left(\frac{4\pi R_{s}(f_{c}+f_{ \tau})}{c}\sqrt{1+\frac{c^{2}f_{\eta}^{2}}{8(f_{c}+f_{\tau})^{2}Lr\beta_{1} \omega_{s}^{2}}}+\frac{\pi f_{\tau}^{2}}{K_{r}}\right)\right), \tag{28}\]  
where \(W_{3}=A\sigma_{r}W_{r}(f_{\tau})W_{a}(f_{\eta}-f_{\eta_{c}})\); here, \(W_{a}(f_{\eta}-f_{\eta_{c}})\) denotes the envelope of the azimuth spectrum, which is centered at the Doppler centroid frequency \(f_{\eta_{c}}\).