## 3 Workflow and Validity Analysis of HDE Algorithm
### Effect of approximation on deceptive jamming Imaging
\begin{split}& H(\xi,\eta)=\text{rect}\bigg{(}\frac{\eta}{bcT_{r}} \bigg{)}w^{2}\bigg{(}\frac{\xi}{2\bar{\Omega}_{x}}\bigg{)}\sigma(x_{P},r_{P}) \exp\bigg{\{}-j\bigg{(}\eta+\frac{4\pi}{\lambda}\bigg{)}r_{P}+j\eta\delta e_{P }(x_{P})\bigg{\}}\exp\Bigg{\{}j\Bigg{[}\frac{\ddot{\delta}e_{P}(x_{P})}{\big{(} 1-2r_{P}\ddot{\delta}e_{P}(x_{P})\big{)}}\frac{r_{P}^{2}}{\big{(}\eta+\frac{4 \pi}{\lambda}\big{)}}\xi^{2}-x_{P}\xi\Bigg{]}\Bigg{\}}\\ &=\text{rect}\bigg{(}\frac{\eta}{bcT_{r}}\bigg{)}w^{2}\bigg{(} \frac{\xi}{2\bar{\Omega}_{x}}\bigg{)}\sigma(x_{P},r_{P})\exp\bigg{\{}-j\bigg{(} \eta+\frac{4\pi}{\lambda}\bigg{)}r_{P}+j\eta\delta e_{P}(x_{P})\bigg{\}}\exp \Bigg{\{}j\Bigg{[}\frac{r_{P}\ddot{\delta}e_{P}(x_{P})}{\big{(}1-2r_{P}\ddot{ \delta}e_{P}(x_{P})\big{)}}\frac{\lambda}{4\pi}\bigg{(}1-\frac{4\eta}{4\pi} \bigg{)}r_{P}\xi^{2}-x_{P}\xi\Bigg{]}\bigg{\}},\end{split}  \tag{48}\]  
where \(\bar{\Omega}_{x}=\Omega_{x}\cdot\big{(}1-2r_{P}\ddot{\delta}e_{P}(x_{P})\big{)}\). Since the range-azimuth coupling factor \(\exp\Big{(}-j\frac{\xi^{2}}{2}\frac{\lambda r_{P}}{4\pi}\frac{\eta\lambda}{4\pi} \Big{)}\) is approximately ignored in the two-dimensional frequency domain in the derivation of Equation (30) to Equation (32), the jamming signal spectrum becomes:  
\[ \begin{split}& H(\xi,\eta)=\text{rect}\bigg{(}\frac{\eta}{bcT_{r}} \bigg{)}w^{2}\bigg{(}\frac{\xi}{2\bar{\Omega}_{x}}\bigg{)}\sigma(x_{P},r_{P}) \times\exp\bigg{\{}-j\bigg{(}\eta+\frac{4\pi}{\lambda}\bigg{)}r_{P}+j\eta \delta e_{P}(x_{P})\bigg{\}}\\ &\times\exp\Bigg{\{}j\Bigg{[}\frac{\lambda r_{P}}{4\pi}\xi^{2} \frac{r_{P}\ddot{\delta}e_{P}(x_{P})}{\big{(}1-2r_{P}\ddot{\delta}e_{P}(x_{P}) \big{)}}-\frac{\lambda r_{P}}{4\pi}\frac{\eta\lambda}{4\pi}\frac{\xi^{2}}{2} \frac{4r_{P}\ddot{\delta}e_{P}(x_{P})-1}{\big{(}1-2r_{P}\ddot{\delta}e_{P}(x_{P} )\big{)}}-x_{P}\xi\Bigg{]}\Bigg{\}}.\end{split}  \tag{49}\]  
The range compression result of the jamming signal can be obtained by performing the range IFT on the above equation:  
\[ \begin{split}& H\big{(}\xi,r^{\prime}\big{)}=\sin c\bigg{[}\frac{ \Omega_{r}}{\pi}\big{(}r^{\prime}-r_{P}-\Delta RCM\big{)}\bigg{]}\sigma(x_{P},r_{P} )\\ & w^{2}\bigg{(}\frac{\xi}{2\bar{\Omega}_{x}}\bigg{)}\times\exp \Bigg{\{}-j\Bigg{[}\frac{4\pi}{\lambda}r_{P}+x_{P}\xi-\frac{\lambda r_{P}}{4 \pi}\xi^{2}\frac{r_{P}\ddot{\delta}e_{P}(x_{P})}{\big{(}1-2r_{P}\ddot{\delta}e_ {P}(x_{P})\big{)}}\Bigg{]}\Bigg{\}},\end{split}  \tag{50}\]where \(\Delta\) RCM is the residual RCM:  
\[\Delta\,\text{RCM}=\frac{\lambda r_{p}}{4\pi}\frac{\lambda}{4\pi}\frac{\xi^{2}}{ 2}\frac{4r_{p}\bar{\delta}e_{p}(x_{p})-1}{\left(1-2r_{p}\bar{\delta}e_{p}(x_{p}) \right)}-\delta e_{p}(x_{p}). \tag{51}\]  
The residual RCM will cause the signal energy to be spread in multiple range gates in the range direction, resulting in the broadening of the main lobe. The decrease of Doppler bandwidth at each range gate also causes a broadening of the main lobe in azimuth. The residual RCM has an error bound: for range and azimuth both broadening of less than 2% due to residual RCM error, the residual range migration should be less than 0.5 range resolution cells [1]. By substituting Equation (45) into Equation (51), the limitation of the maximum trajectory deviation \(d_{max}\) according to the residual RCM can be obtained as follows:  
\[d_{max}<\frac{cR_{0}}{4B_{r}\Delta r_{P}}-\frac{r_{P}}{8\Delta r_{P}}\frac{L_{ a}^{2}}{R_{0}}. \tag{52}\]  
The residual RCM analysis of the HDE algorithm against different SAR maximum trajectory offsets, jamming range swath widths, and flight heights is described in Figure 5.  
We can find that the smaller the SAR trajectory deviation and the lower the flight altitude and resolution, the easier it is for the jamming signal based on the HDE algorithm to meet the limitation of residual RCM after SAR imaging. Next, the azimuth compression results are analyzed, only considering the exponential term in Equation (50).