## 4 Simulation and Result
### Computational Complexity Analysis  
In this section, the computational complexity of the HDE algorithm will be estimated to evaluate its practical value. For ease of analysis, those operations used in the algorithm in Section 2.3.2, such as addition, multiplication, and square root, are considered basic operations that can be performed by the floating-point unit in one instruction cycle. We assume that the jamming scene template consists of \(M\times N\) point scatterers, where the number of azimuth cells is \(M\) and the number of range cells is \(N\). The following part will analyze the number of basic operations of jamming modulation under a specific number of range sampling points \(N_{r}\) and azimuth sampling points \(N_{a}\). Figure 14: Partial enlargement in red boxes. (**a**) Actual scene generated using RS; (**b**) Fake scene generated using HDE. Figure 15: Fake scene deceptive jamming simulation. (**a**) Imaging result of the actual scene echo, parts of the actual image marked by rectangular red box; (**b**) Imaging result after adding the jamming signal to the echo by HDE, parts of the jamming image marked by rectangular red box. In the initialization stage, for a scattering point \(\sigma(x_{P},r_{P})\), calculating the slant range \(r_{P}\) requires seven basic operations. Since the complex number calculation completes the nominal slant range \(r_{P}\) phase compensation, the basic operation amount is 5, and the azimuth FFT requires \(5N_{a}log_{2}N_{a}\) basic operations. Therefore, the basic operation of \(12 MN+5N_{a}N_{r}log_{2}(N_{a})\) is needed to complete the azimuth time-frequency conversion of the deceptive jamming template. \(G_{A}(\xi,r_{P})\) requires \(8 MN\) basic operations. Therefore, combined with the azimuth inverse Fourier transform, the basic operation amount in the initialization stage is:  
\[C_{\text{initialization}}=10N_{a}N_{r}\log_{2}(N_{a})+20 MN. \tag{58}\]  
In the real-time calculation stage, the components of the trajectory deviation need to be calculated in each PRI period, and the basic operation amount of \(\delta r(x^{\prime})\) and \(\psi(x^{\prime},r_{P})\) is \(7N_{a}+3N_{r}\). Completing the construction of \(\delta r(x^{\prime})\) and \(\psi(x^{\prime},r_{P})\) is \(7N_{a}+3N_{r}\) requires a total of \(12N_{a}N_{r}\) basic operations, and the range FFT requires \(5N_{r}log_{2}N_{r}\) basic operations. Therefore, the basic operations of \(5N_{r}\log_{2}N_{r}+15N_{r}+7N_{a}\) are required to complete the JSF operation of a PRI in the real-time computation stage, and the basic operation amount of convolutional forwarding modulation combined with DRFM is:  
\[C_{\text{real-time}}=15N_{r}\log_{2}(N_{r})+15N_{r}+7N_{a}. \tag{59}\]  
From the above analysis, the characteristics of the HDE algorithm can be summarized. The main calculation amount of the HDE algorithm is the Fourier transform, whether it is the initialization stage or the real-time calculation stage. In addition, the construction for nonlinear-trajectory in the real-time stage can be accomplished simply by complex multiplication. Similarly, the basic operation amount of the traditional deceptive jamming SA algorithm for nonlinear-trajectory SAR is analyzed, which is listed in Table 3.  
If the nonlinear-trajectory jamming modulation characteristics are not discussed, only the calculation efficiency of the algorithm is considered. In Table 3, the basic operation amount of the linear trajectory SAR efficient deceptive jamming algorithm, the spatial frequency-domain interpolation (SFI) [23] algorithm and the segmented modulation (SM) [14] algorithm are supplemented. To observe the comparison between the computational complexity of these algorithms more clearly, control \(N_{a}=N_{r}=M=N\), and draw the logarithmic graph of each algorithm's total and real-time operation amount, as shown in Figure 16. It can be seen that the operation amount of the HDE algorithm is lower than the SA algorithm in the whole jamming modulation process or the real-time calculation stage.