## 3 Workflow and Validity Analysis of HDE Algorithm
### HDE Algorithm Workflow  
In the workflow of the jammer, obtaining the relevant parameters of the jamming object is the premise of jamming, which mainly includes the following aspects:  
* SAR platform parameters, such as SAR flight altitude \(H\), motion velocity \(v\);
* Antenna parameters, such as antenna elevation angle \(\theta_{0}\), synthetic aperture length \(L\);
* Signal parameters, such as the carrier frequency \(f\), bandwidth \(\Delta f\), chirp pulse duration \(T_{r}\) and PRI \(T_{p}\);  
The above parameters need to be obtained before jamming modulation to estimate the SAR nominal linear trajectory and construct the jamming frequency response function of the SAR nominal linear trajectory. When the jamming object is a nonlinear-trajectory airborne SAR, the trajectory deviation \(\mathbf{d}\) of the SAR also needs to be obtained in real time for the trajectory deviation construction of the HDE algorithm. The specific acquisition methods of these parameters will not be discussed in this paper, and we assume that the parameters have been obtained in advance. The main workflow of the deceptive jamming algorithm for nonlinear-trajectory SAR based on HDE, as per Equation (36), is illustrated in Figure 4. The workflow of the HDE algorithm encompasses two stages: jammer initialization and real-time computation. 1. The first stage is jammer initialization. First, a deceptive template with \(\sigma(x_{P},r_{P})\) complex backscatter coefficients is obtained. Then according to the SAR platform parameters and signal parameters, the jammer performs the phase compensation of the shortest slant range \(r_{P}\) on the deceptive template and combines the antenna parameters to construct the azimuth frequency modulation term of the deceptive template in the SAR azimuth frequency domain. Finally, the azimuth frequency modulation term is converted to the azimuth time domain to form an initialization template for the jammer to perform trajectory deviation construction in the real-time calculation stage. 2. In the real-time calculation stage, since the trajectory deviation of nonlinear-trajectory SAR is difficult to calculate in advance, the jammer needs to obtain the SAR trajectory deviation in real time for pulse-by-pulse trajectory deviation construction. First, the SAR trajectory deviation \(\mathbf{d}\) of the current pulse time is obtained, and the trajectory deviation components \(\delta r(x^{\prime})\) and \(\psi(x^{\prime},r_{P})\) are calculated through Equation (5). Then the jammer performs the construction of the range space-variant trajectory deviation \(\psi(x^{\prime},r_{P})\) in the range time domain, and performs the construction of the  
Figure 4: Hybrid domain efficient (HDE) deceptive jamming algorithm workflow. space-invariant trajectory deviation \(\delta r(x^{\prime})\) and the range frequency modulation in the range frequency domain, from which the current pulse SAR system-related filter \(H_{trans,P}(x^{\prime},\eta)\) can be constructed. Finally, the jammer obtains the JFR \(H_{jammer,P}(x^{\prime},\eta)\) for the current pulse time based on Equation (36). After generating the initialization template based on the nominal linear trajectory parameters of SAR, the jammer needs to repeat real-time calculations to construct for trajectory deviation following every pulse interception. This iterative process ultimately results in the deceptive jamming of nonlinear-trajectory SAR. For the HDE algorithm, whether in the initialization stage of the jammer or the real-time calculation stage, the calculation is completed only by Fourier transform and complex multiplication operations, which have low computational complexity. Furthermore, compared with the existing deceptive jamming algorithms in the azimuth time domain and azimuth frequency domain, the algorithm proposed in this paper adopts a new idea of azimuth hybrid domain modulation. The algorithm integrates the high-precision trajectory deviation model into the jamming modulation.