## 2 Nonlinear-Trajectory SAR Deception Jamming Algorithm Based on HDE
### Nonlinear-Trajectory SAR System Model  
Taking side-looking Stripmap SAR as an example, when the jammer interferes with the nonlinear-trajectory SAR system, the geometric relationship between the SAR, the jammer, and the deceptive area is shown in Figure 1. Where the origin \(O\) of the spatial coordinate system is located at the flight start point of the SAR radar, \(X^{\prime}\) is the trajectory of the radar. The X-axis points to the azimuth direction, the Y-axis points to the range direction, and the Z-axis points to the height direction. Vector \(d(x^{\prime})\) represents the absolute deviation vector of the nonlinear-trajectory relative to the nominal trajectory. The \(y\) and \(z\) components of vector \(d(x^{\prime})\) represent the horizontal and vertical deviation of the SAR flight platform relative to the nominal trajectory. \(\beta\) represents the angle of the flight trajectory deviation relative to the nominal trajectory, and \(\theta_{0}\) is the SAR antenna elevation angle. The jammer is placed at point \(J(x_{j},r_{j},\theta_{j})\), \(P(x_{p},r_{p},\theta_{p})\) represents an arbitrary point in the deceptive region \(P\), and the point on the center of the beam with the same nearest slant range and height as it is \(P_{d}(x^{\prime},r_{p},\theta_{p})\). \(\theta_{p}\) and \(\theta_{j}\) are the local look angles of points \(P\) and \(J\). \(r_{P}\) is the shortest slant range between the radar antenna and point \(P\) in the nominal trajectory, \(R_{p}^{\prime}\) and \(R_{P}\) are the slant range from the target \(P\) to the antenna in the generic antenna position for nonlinear and nominal trajectories, \(\Delta r_{P}\) is the ground range offset of point \(P\) with respect to the scene centroid, respectively. Figure 1: Nonlinear trajectory SAR spatial geometry model. (**a**) Three-dimension view; (**b**) Two-dimension cross trajectory view. Assume that the SAR transmitted signal is a chirp pulse, as follows:  
\[p(t)=\text{rect}\bigg{(}\frac{t}{T_{r}}\bigg{)}\exp\Big{(}j2\pi ft-j\pi K_{r}t^{2} \Big{)}, \tag{1}\]  
where \(t\) is the fast time, \(f\) is the carrier frequency, and \(T_{r}\) and \(K_{r}\) are the chirp pulse duration and rate, respectively. The spatial domain expression of the nonlinear-trajectory SAR echo signal in the deceptive area \(P\) is [33]:  
\[h\big{(}x^{\prime},r^{\prime}\big{)}=\text{rect}\bigg{(}\frac{r^{\prime}-R_{P} ^{\prime}}{cT_{r}/2}\bigg{)}w^{2}\bigg{(}\frac{x^{\prime}-x_{P}}{L_{a}}\bigg{)} \iint\sigma(x_{P},r_{P})\exp\bigg{[}-j\frac{4\pi}{\lambda}R_{P}^{\prime}-j \frac{4\pi}{\lambda}\frac{\Delta f/f}{cT_{r}}\big{(}r^{\prime}-R_{P}^{\prime} \big{)}^{2}\bigg{]}dxdr, \tag{2}\]  
where \(x^{\prime}\) is the actual position of SAR radar, \(r^{\prime}=ct/2\) is the range spatial sampling interval and \(c\) is the speed of light. \(w(\cdot)\) is the antenna ground illumination pattern, \(L_{a}=\lambda r/L\) is the synthetic aperture length, and \(L\) is the azimuth length of the physical antenna. \(\sigma(x_{P},r_{P})\) is the target reflectivity. \(\lambda\) is the carrier wavelength, \(\Delta f=K_{r}T_{r}/2\pi\) is the signal bandwidth. From the geometry shown in Figure 1, the slant ranges \(R_{P}^{\prime}\) and \(R_{P}\) take the forms:  
\[R_{P}=\sqrt{r_{P}^{2}+(x^{\prime}-x_{P})^{2}}, \tag{3}\]  
\[R_{P}^{\prime}=\sqrt{\big{(}r_{P}+\delta r_{r}(x^{\prime},r_{P})\big{)}^{2}+( x^{\prime}-x_{P})^{2}}. \tag{4}\]  
The geometry from cross trajectory view is shown in Figure 1, so the term \(\delta r_{r}(x^{\prime},r_{P})\) is given by:  
\[\delta r_{r}\big{(}x^{\prime},r_{P}\big{)}=\sqrt{r_{P}^{2}+d(x^{ \prime})^{2}-2d(x^{\prime})r_{P}\sin(\theta_{P}-\beta(x^{\prime}))}-r_{P} \tag{5}\] \[\approx-d\big{(}x^{\prime}\big{)}\sin\big{[}\theta_{P}-\beta(x^{ \prime})\big{]},\]  
where \(d(x^{\prime})=|\mathbf{d}|\) are related to the horizontal and vertical trajectory deviation of the SAR radar.