## 2 Nonlinear-Trajectory SAR Deception Jamming Algorithm Based on HDE
### Nonlinear-Trajectory SAR Deception Jamming Principle  
First, we illustrate the geometric relationship between jammer, deceptive area, and SAR by establishing a coordinate system: SAR radar coordinates and jammer coordinates in the two-dimensional slant-range plane, as shown in Figure 2. The dashed line on the \(x^{\prime}\) axis corresponds to the nominal linear trajectory of the SAR radar, and the curve on the \(x^{\prime}\) axis is the actual nonlinear-trajectory of the SAR radar. The spatial origin \(O_{I}\) is at jammer.  
Then the jammer is located at \((r_{J},0)\) in the coordinate system, \(r_{J}\) represents the shortest slant range between the jammer and the radar. Denote location of an arbitrary false target by \((r_{P},x_{P})\) in deceptive area, and its scattering coefficient is \(\sigma(x_{P},r_{P})\), \(R^{\prime}_{p}(x^{\prime},x_{P},r_{P})\) and \(R^{\prime}_{J}(x^{\prime},x_{J},r_{J})\) represent the instantaneous slant range of false target \(P\) and jammer \(J\) at the SAR radar azimuth position \(x^{\prime}\), respectively.  
An implementation scheme of SAR deceptive jamming based on modulation-retransmission is presented in Figure 3. The jammer needs amplification, down-conversion, analog to digital (A/D) conversion, and Fourier transform of the intercepted SAR radar signal to obtain the baseband range frequency domain signal, multiply it with the jamming frequency response (JFR) and then perform inverse Fourier transform, digital to analog (D/A) conversion, up-conversion and gain control to generate the jamming signal and forward it to the SAR radar. Therefore, the JFR is the key to deceptive jamming modulation. Equation (2) is expressed in the range frequency domain - form by the stationary phase method:  
\[\begin{split} H\big{(}x^{\prime},\eta\big{)}&= \text{rect}\bigg{(}\frac{\eta}{bcT_{r}}\bigg{)}\exp\bigg{(}j\frac{\eta^{2}}{4b} \bigg{)}\\ &\quad\times\iint\sigma(x_{P},r_{P})w^{2}\bigg{(}\frac{x^{\prime }-x_{P}}{L_{a}}\bigg{)}\exp\bigg{[}-j\bigg{(}\eta+\frac{4\pi}{\lambda}\bigg{)} R^{\prime}_{P}\bigg{]}dxdr,\end{split} \tag{9}\]  
where \(b=\frac{4\pi K_{r}}{c^{2}}\). \(\eta=4\pi f_{r}/c\) represents range frequency. The double integral term is the transfer function of the SAR system. Equation (9) combined with Figure 2, the JFR can be obtained as:  
\[H_{jammer,P}\big{(}x^{\prime},\eta\big{)}=\text{rect}\bigg{(}\frac{\eta}{bcT_{ r}}\bigg{)}\exp\bigg{(}j\frac{\eta^{2}}{4b}\bigg{)}\iint\sigma(x_{P},r_{P}) \frac{w^{2}\bigg{(}\frac{x^{\prime}-x_{P}}{L_{a}}\bigg{)}}{w^{2}\bigg{(}\frac {x^{\prime}-x_{J}}{L_{a}}\bigg{)}}\exp\bigg{[}-j\bigg{(}\eta+\frac{4\pi}{ \lambda}\bigg{)}\big{(}\Delta R^{\prime}_{P}\big{(}x^{\prime},x_{P},r_{P} \big{)}\big{)}\bigg{]}dxdr, \tag{10}\]  
where \(\Delta R^{\prime}_{p}\big{(}x^{\prime},x_{P},y_{P}\big{)}\) is the instantaneous slant range between \(R^{\prime}_{P}\) and \(R^{\prime}_{J}\):  
\[\begin{split}\Delta R^{\prime}_{P}\big{(}x^{\prime},x_{P},r_{P} \big{)}=R^{\prime}_{P}-R^{\prime}_{J}&=\sqrt{\left[r_{P}+\delta r _{r}(x^{\prime},r_{P})\right]^{2}+\left(x^{\prime}-x_{P}\right)^{2}}\\ &\quad-\sqrt{\left[r_{J}+\delta r_{r}\big{(}x^{\prime},r_{J}\big{)} \right]^{2}+\left(x^{\prime}-x_{J}\right)^{2}}.\end{split} \tag{11}\]  
Equation (10) is the JFR of the traditional point-by-point superposition direct calculation algorithm (SA) [11]. The jammer calculates different JFRs in each pulse according to  
Figure 2: Two-dimension slant range plane geometric model for SAR deceptive jamming.  
Equation (10), which can achieve jamming modulation of the deceptive template. Due to the dependence of nonlinear-trajectory deceptive jamming on real-time trajectory deviation, the jammer needs to update the JFR for each PRI. However, the SA algorithm involves double integration calculations under continuous pulses, which results in a large computation load and makes it difficult to ensure the flexible and efficient calculation of JFR. Therefore, it is necessary to calculate the JFR more efficiently and flexibly.