## 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
However, \(G(\xi,\eta,r_{P})\) can be decomposed into the \(\eta\)-dependent part \(G_{B}(\xi,\eta,r_{P})\) and the remaining part \(G_{A}(\xi,r_{P})\)[34]:  
\[G(\xi,\eta,r_{P})=\text{rect}\bigg{(}\frac{\eta}{bcT_{r}}\bigg{)}w^{2}\bigg{(} \frac{\xi}{\Omega_{x}}\bigg{)}\exp\left[-j\bigg{(}\sqrt{\bar{\eta}^{2}-\xi^{2} }-\bar{\eta}\bigg{)}r_{P}\right], \tag{18}\]1  
Here,  
\[ \begin{split} G_{A}(\xi,r_{P})=w\bigg{(}\frac{\xi}{\Omega_{x}} \bigg{)}\exp\Bigg{\{}-j\bigg{[}\sqrt{\bigg{(}\frac{4\pi}{\lambda}\bigg{)}^{2} -\xi^{2}}-\frac{4\pi}{\lambda}\bigg{]}r_{P}\Bigg{\}}\\ G_{B}(\xi,\eta,r_{P})=\text{rect}\bigg{(}\frac{\eta}{bcT_{r}} \bigg{)}w\bigg{(}\frac{\xi}{\Omega_{x}}\bigg{)}\exp\bigg{(}-j\frac{\xi^{2}}{2 }\frac{\lambda r_{P}}{4\pi}\frac{\eta\lambda}{4\pi}\bigg{)}.\end{split}  \tag{29}\]  
When \(\Psi(\xi,r_{P})\) the spectral expansion along the \(\xi\) axis is much more minor than the minimum period of oscillation of the \(G_{B}(\xi,\eta,r_{P})\), the \(G_{B}(\xi,\eta,r_{P})\) approximates that it does not vary along the \(\xi\) axis within the \(\xi\) bandwidth of the \(\Psi(\xi,r_{P})\). So that \(G_{B}(\xi,\eta,r_{P})\) can be separated from the convolution calculation along the \(\xi\) (this assumption will be analyzed in the next section). It means the convolution operation in the azimuth frequency domain only needs to be calculated once:  
\[\bar{H}(x^{\prime},\eta)=\int\exp\bigg{[}-j\bigg{(}\eta+\frac{4\pi}{\lambda} \bigg{)}r_{P}\bigg{]}dr\int G_{B}(\xi,\eta,r_{P})\bigg{\{}[G_{A}(\xi,r_{P}) \Gamma(\xi,r_{P})]\underset{\xi}{\otimes}\Psi(\xi,r_{P})\bigg{\}}\exp\big{(} j\xi x^{\prime}\big{)}d\xi. \tag{30}\]  
Since \(\frac{\eta\lambda}{4\pi}\ll 1\), we can try to neglect the range-azimuth coupling phase in \(G_{B}(\xi,\eta,r_{P})\), and its effect on the jamming signal will be analyzed in Section 3.2. Then \(\bar{H}(x^{\prime},\eta)\) can be simplified to the following form:\[\bar{H}(x^{\prime},\eta)\approx G_{B}(\eta)\int\exp\bigg{[}-j\bigg{(}\eta+\frac{4 \pi}{\lambda}\bigg{)}r_{p}\bigg{]}dr\int\bigg{\{}[G_{A}(\xi,r_{p})\Gamma(\xi,r)] \otimes\Psi(\xi,r_{p})\bigg{\}}\exp(j\xi x^{\prime})d\xi. \tag{31}\]  
Here,  
\[G_{B}(\eta)=\text{rect}\bigg{(}\frac{\eta}{bcT_{r}}\bigg{)}. \tag{32}\]  
The last integral term of Equation (31) represents the inverse Fourier transform of the convolution of the two azimuthal frequency domain terms \(\Psi(\xi,r_{p})\) and \(G_{A}(\xi,r_{p})\Gamma(\xi,r)\), which can be converted into the product of the azimuthal time-domain terms:  
\[\bar{H}(x^{\prime},\eta)=G_{B}(\eta)\int\exp\bigg{[}-j\bigg{(}\eta+\frac{4\pi} {\lambda}\bigg{)}r_{p}\bigg{]}\exp\bigg{[}-j\frac{4\pi}{\lambda}\psi(x^{ \prime},r_{p})\bigg{]}IFT_{\xi}[G_{A}(\xi,r_{p})\Gamma(\xi,r_{p})]dr. \tag{33}\]  
Rearrange the integral equation as follows, where \(FT_{r}\) represents the range Fourier transform and \(IFT_{\xi}\) represents the azimuth inverse Fourier transform:  
\[\bar{H}(x^{\prime},\eta)=G_{B}(\eta)FT_{r}\bigg{\{}\exp\bigg{[}-j\frac{4\pi}{ \lambda}\psi\big{(}x^{\prime},r_{p}\big{)}\bigg{]}IFT_{\xi}\bigg{[}G_{A}(\xi, r_{p})FT_{\chi}\bigg{(}\exp(-\frac{4\pi}{\lambda}r_{p})\sigma(x_{p},r_{p}) \bigg{)}\bigg{]}\bigg{\}}. \tag{34}\]  
Combined with Equation (15), the SAR system-related filter is as follows:  
\[ \begin{split} H_{trans,p}(x^{\prime},\eta)=&\exp \bigg{[}-j\bigg{(}\eta+\frac{4\pi}{\lambda}\bigg{)}\delta r(x^{\prime})\bigg{]} G_{B}(\eta)\\ &\times FT_{r}\bigg{\{}\exp\bigg{[}-j\frac{4\pi}{\lambda}\psi \big{(}x^{\prime},r_{p}\big{)}\bigg{]}IFT_{\xi}\bigg{[}G_{A}(\xi,r_{p})FT_{ \chi}\bigg{(}\exp(-\frac{4\pi}{\lambda}r_{p})\sigma(x_{p},r_{p})\bigg{)} \bigg{]}\bigg{\}}.\end{split}  \tag{35}\]  
Equation (35) realizes the double integral calculation of the deceptive template efficiently in the azimuth frequency domain through azimuth Fourier transform and complex multiplication, and moves the trajectory deviation term to the azimuth time domain.