## 2 Nonlinear-Trajectory SAR Deception Jamming Algorithm Based on HDE
### Deceptive Jamming Based on HDE
#### 2.3.2 Fast Algorithm of the SAR System-Related Filter Based on HDE
Transform the time-domain term multiplication in the azimuth Fourier transform into a frequency domain term convolution, rearrange its integral expression [33]:  
\[\bar{H}(\xi,\eta)=\int\exp\left[-j\bigg{(}\eta+\frac{4\pi}{\lambda}\bigg{)}r_ {P}\right]dr\bigg{\{}[G(\xi,\eta,r_{P})exp(-j\xi x)]\underset{\xi}{\otimes}F( \xi,\eta,r_{P})\bigg{\}}, \tag{17}\]  
where \(G(\xi,\eta,r_{P})\) represents the frequency response function of the nominal linear trajectory SAR, and \(F(\xi,\eta,r_{P})\) represents the ground reflectivity term corrected by the space-variant trajectory deviation:  
\[G(\xi,\eta,r_{P})=\text{rect}\bigg{(}\frac{\eta}{bcT_{r}}\bigg{)}w^{2}\bigg{(} \frac{\xi}{\Omega_{x}}\bigg{)}\exp\left[-j\bigg{(}\sqrt{\bar{\eta}^{2}-\xi^{2} }-\bar{\eta}\bigg{)}r_{P}\right], \tag{18}\]  
\[F(\xi,\eta,r_{P})=\iint f\big{(}x_{P},x^{\prime},\eta,r_{P}\big{)}\exp\left(- j\xi x^{\prime}\big{)}dxdx^{\prime}, \tag{19}\]  
\[f\big{(}x_{P},x^{\prime},\eta,r_{P}\big{)}=\sigma(x_{P},r_{P})\exp\big{\{}-j \bar{\eta}\big{[}\psi\big{(}x^{\prime},r_{P}\big{)}+\varphi\big{(}x^{\prime},x _{P},r_{P}\big{)}\big{]}\big{\}}. \tag{20}\]  
Here,  
\[\Omega_{x}=\frac{2\pi L_{a}}{\lambda R_{0}},\quad\bar{\eta}=\eta+\frac{4\pi}{ \lambda}. \tag{21}\]  
To facilitate the real-time trajectory deviation construction of the jammer, Equation (17) is transformed into the azimuth time domain for the derivation of the trajectory deviation:  
\[\bar{H}\big{(}x^{\prime},\eta\big{)}=\int\exp[-j\bar{\eta}r_{P}]dr\int\exp \big{(}j\xi x^{\prime}\big{)}d\xi\bigg{\{}[G(\xi,\eta,r_{P})exp(-j\xi x)] \underset{\xi}{\otimes}F(\xi,\eta,r_{P})\bigg{\}}. \tag{22}\]  
The center beam approximation is assumed in Equation (20), which takes that the motion error associated with all the targets' positions in the interior of the azimuth beam is equal to the motion error at the center of the azimuth beam. In addition, considering that the range frequency window function in \(G(\xi,\eta,r_{P})\) has a limit \(|\eta|<b(cT_{r}/2)\), there is \(\eta\ll\frac{4\pi}{\lambda}\). Then (20) can be expressed as:  
\[f\big{(}x_{P},x^{\prime},\eta,r_{P}\big{)}\approx\sigma(x_{P},r_{P})\exp\bigg{\{} -j\bigg{(}\frac{4\pi}{\lambda}\bigg{)}\psi\big{(}x^{\prime},r_{P}\big{)}\bigg{\}}. \tag{23}\]  
According to the definition of SAR motion compensation [29], the effect of the assumption on imaging can be ignored when the maximum phase error caused by this assumption is much less than 1 rad. The following conditions are met:  
\[\bigg{|}\bigg{(}\eta+\frac{4\pi}{\lambda}\bigg{)}\varphi\big{(}x^{\prime},x_{P },r_{P}\big{)}+\eta\psi\big{(}x^{\prime},r_{P}\big{)}\bigg{|}\ll 1\quad. \tag{24}\]  
Then Equation (22) can be simplified to the following form [34]:  
\[\bar{H}\big{(}x^{\prime},\eta\big{)}=\int\exp\bigg{[}-j\bigg{(}\eta+\frac{4 \pi}{\lambda}\bigg{)}r_{P}\bigg{]}dr\int\exp\big{(}j\xi x^{\prime}\big{)}d \xi\bigg{\{}[G(\xi,\eta,r_{P})\Gamma(\xi,r_{P})]\underset{\xi}{\otimes}\Psi( \xi,r_{P})\bigg{\}}. \tag{25}\]  
Here, both \(FT_{x}\) and \(FT_{x^{\prime}}\) represent azimuth Fourier transform:  
\[\Gamma(\xi,r_{P})=FT_{x}\big{\{}\sigma(x_{P},r_{P})\big{\}}, \tag{26}\]  
\[G(\xi,\eta,r_{P})=\text{rect}\bigg{(}\frac{\eta}{bcT_{r}}\bigg{)}w^{2}\bigg{(} \frac{\xi}{\Omega_{x}}\bigg{)}\exp\left[-j\bigg{(}\sqrt{\bar{\eta}^{2}-\xi^{2} }-\bar{\eta}\bigg{)}r_{P}\right], \tag{18}\]0  
The meaning of the last integral terms in Equation (25) is to represent the inverse Fourier transform of the convolution of the deviation phase component of the range space-variant trajectory \(\Psi(\xi,r_{P})\) and SAR system frequency response function \(G(\xi,\eta,r_{P})\Gamma(\xi,r_{P})\). Due to the dependence of the \(G(\xi,\eta,r_{P})\) function on the \(\eta\) item, for a JFR with N points sampled in the range domain, the convolution operation needs to be calculated N times. Such modulation has a high computational time complexity.