## 3 Workflow and Validity Analysis of HDE Algorithm
### Effect of approximation on deceptive jamming Imaging  
In this subsection, the SAR motion compensation imaging analysis is performed on the jamming signal of the HDE deceptive jamming algorithm. According to the validity analysis of the above approximation conditions, the influence of these approximations on the imaging results and the effective area of deceptive jamming are analyzed in detail. Taking the jamming of \(P\) point as an example, the jamming signal with the residual error of motion construction is equivalently expressed in \((x^{\prime},\eta)\) domain as follows:  
\[ \begin{split} H\big{(}x^{\prime},\eta\big{)}=& \text{rect}\bigg{(}\frac{\eta}{bcT_{r}}\bigg{)}w^{2}\bigg{(}\frac{x^{ \prime}-x_{P}}{L_{a}}\bigg{)}\sigma(x_{P},r_{P})\times\\ &\exp\bigg{\{}-j\bigg{(}\eta+\frac{4\pi}{\lambda}\bigg{)}\bigg{[} r_{P}+\frac{(x^{\prime}-x_{P})^{2}}{2r_{P}}-\varphi\big{(}x^{\prime},x,r \big{)}\bigg{]}+j\eta\psi\big{(}x^{\prime},r\big{)}\bigg{\}}.\end{split}  \tag{44}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameter** & **Transmitted** & **Carrier** & & & & **Platform** & **Antenna** \\
**Radar Signal** & **Frequency** & **Chirp Rate** & **PRI** & **Pulse Width** & **Velocity** & **Aperture** \\ \hline Value & LFM & 10 GHz & 15 MHz/ μs & 5.6 ms & 10 μs & 150 m/s & 2 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The setting of SAR parameters in simulations. Through Equations (7) and (8), the residual error can be expressed as:  
\[ \begin{split}&\delta e_{P}\big{(}x^{\prime}\big{)}=\psi\big{(}x^{ \prime},r\big{)}+\varphi\big{(}x^{\prime},x,r\big{)}=\psi\big{(}x^{\prime},r \big{)}+\frac{(\delta r(x^{\prime})+\psi(x^{\prime},r))}{2r_{P}^{2}}\big{(}x^{ \prime}-x_{P}\big{)}^{2}\\ &=\delta e_{P}(x_{P})+\ddot{\delta}e_{P}(x_{P})\cdot\big{(}x^{ \prime}-x_{P}\big{)}^{2}.\end{split}  \tag{45}\]  
where \(\psi(x^{\prime},r)\) can be equivalent to a fixed error term, and \(\varphi(x^{\prime},x,r)\) can be equivalent to a quadratic error term. By substituting Equation (45) into Equation (44) and making azimuth FT, the two-dimensional frequency domain form of the jamming signal can be obtained:  
\[ \begin{split} H(\xi,\eta)=&\text{rect}\bigg{(} \frac{\eta}{bcT_{r}}\bigg{)}\int w^{2}\bigg{(}\frac{x^{\prime}-x_{P}}{L_{a}} \bigg{)}\sigma(x_{P},r_{P})\\ &\times\exp\bigg{\{}-j\bigg{(}\eta+\frac{4\pi}{\lambda}\bigg{)} \Bigg{[}r_{P}+\frac{(x^{\prime}-x_{P})^{2}}{2r_{P}}-\ddot{\delta}e_{P}(x_{P}) \cdot\big{(}x^{\prime}-x_{P}\big{)}^{2}\Bigg{]}+j\eta\delta e_{P}(x_{P}) \bigg{\}}\exp\big{(}-j\xi x^{\prime}\big{)}dx^{\prime}.\end{split}  \tag{46}\]  
The approximate solution of Equation (46) can be obtained by the stationary phase method [1]:  
\[ \begin{split}& H(\xi,\eta)=\text{rect}\bigg{(}\frac{\eta}{bcT_{r}} \bigg{)}w^{2}\Bigg{(}\frac{\xi}{2\Omega_{x}\cdot\big{(}1-2r_{P}\ddot{\delta}e_ {P}(x_{P})\big{)}}\Bigg{)}\sigma(x_{P},r_{P})\\ &\times\exp\bigg{\{}-j\bigg{(}\eta+\frac{4\pi}{\lambda}\bigg{)}r_ {P}+j\eta\delta e_{P}(x_{P})\bigg{\}}\exp\Bigg{\{}j\Bigg{[}\frac{r_{P}^{2}}{ \big{(}\eta+\frac{4\pi}{\lambda}\big{)}\big{(}1-2r_{P}\ddot{\delta}e_{P}(x_{P} )\big{)}}\xi^{2}-x_{P}\xi\Bigg{]}\Bigg{\}}.\end{split}  \tag{47}\]  
The range cell migration and focus depth compensation, and azimuth compression are applied to the above equation, and the Taylor expansion of \(\frac{1}{\big{(}\eta+\frac{4\pi}{\lambda}\big{)}}\) in the last exponential term is carried out to obtain the signal spectrum as follows:  
\[