## 3 Workflow and Validity Analysis of HDE Algorithm
### Effect of approximation on deceptive jamming Imaging
The inverse Fourier transform along the azimuth direction is as follows:  
\[H\big{(}x^{\prime},r^{\prime}\big{)}=\exp\biggl{(}-j\frac{4\pi}{\lambda}r_{P} \biggr{)}\int w^{2}\biggl{(}\frac{\xi}{2\widehat{\Omega}_{x}}\biggr{)}\times \exp\biggl{\{}j\biggl{[}\frac{\lambda r_{P}}{4\pi}\xi^{2}\frac{r_{P}\bar{ \delta}e_{p}(x_{P})}{\left(1-2r_{P}\bar{\delta}e_{p}(x_{P})\right)}\biggr{]} \biggr{\}}\exp\bigl{[}j\xi\big{(}x^{\prime}-x_{P}\big{)}\bigr{]}d\xi, \tag{53}\]  
\[\text{let}\,m(x_{P})=\frac{\lambda r_{P}^{2}}{4\pi}\frac{\bar{\delta}e_{p}(x_ {P})}{\left(1-2r_{p}\bar{\delta}e_{p}(x_{P})\right)},\text{since}\,m(x_{P}) \widehat{\Omega}_{x}{}^{2}\ll 1, \tag{53}\]  
Equation (53) can be equivalent to:  
\[H\big{(}x^{\prime},r^{\prime}\big{)}=\exp\biggl{(}-j\frac{4\pi}{\lambda}r_{P }\biggr{)}\int w^{2}\biggl{(}\frac{\xi}{2\widehat{\Omega}_{x}}\biggr{)}\times \Bigl{(}1+jm(x_{P})\xi^{2}\Bigr{)}\exp\bigl{[}j\xi\big{(}x^{\prime}-x_{P} \big{)}\bigr{]}d\xi. \tag{54}\]  
Figure 5: Due to different jamming range swath widths, the influence of different SAR maximum trajectory offsets and flight altitudes on residual RCM are considered. **(a)** The azimuth resolution is 1 m; **(b)** The azimuth resolution is 2.5 m.  
Solving the integral of Equation (54) and let \(x^{\prime}=x_{P},r^{\prime}=r_{P}\), then [36]:  
\[H\big{(}x^{\prime}=x_{P},r^{\prime}=r_{P}\big{)}=\exp\!\left(-j\frac{4\pi}{ \lambda}r_{P}\right)\cdot\exp\left[j\frac{m(x_{P})\cdot\widehat{\Omega}_{x}^{2 }}{3}\right]. \tag{55}\]  
It can be obtained from Equation (54) that the phase of the jamming signal after imaging is destroyed by the residual quadratic trajectory construction error term \(m(x_{P})\), which will lead to a certain degree of azimuth defocusing and sidelobe asymmetry after azimuth compression. Its influence on the azimuth main lobe broadening can be estimated by the quadratic phase error (QPE). For a typical Kaiser window of \(\beta=2.5\), if the required broadening is less than 2%, the corresponding absolute value of QPE should be less than \(0.27\pi\)[1]. The expression of QPE is as follows:  
\[QPE=\frac{m(x_{P})\cdot\widehat{\Omega}_{x}^{2}}{3}\approx\frac{1}{6}\frac{r_ {P}\pi L_{a}^{2}}{R_{0}^{2}\lambda}\frac{d_{max}}{r_{P}-d_{max}}. \tag{56}\]  
An equivalent substitution of the QPE can be achieved through the limitation of the maximum trajectory deviation \(d_{max}\):  
\[d_{max}<\frac{81r_{P}R_{0}^{2}\lambda}{50r_{P}L_{a}^{2}+81R_{0}^{2}\lambda} \tag{57}\]  
Figure 6 shows the HDE algorithm against different SAR maximum trajectory offsets, azimuth resolution, and jamming range swath widths quadratic phase error. For the SAR with the maximum trajectory deviation of 4 m and the false target within the jamming range width of 8000 m, the residual quadratic trajectory construction phase error is much smaller than \(\lambda/4\pi\), and the resulting defocus is negligible. In summary, the jamming effect after imaging based on the HDE algorithm is mainly affected by the residual RCM in Equation (51). To achieve the desired deceptive jamming effect, the HDE algorithm needs to restrict the SAR flight height, azimuth resolution, and jamming range swath width according to Equation (52), which leads to the fact that the effective jamming area in the range direction does not exceed the jamming swath width. However, there are no restrictions on the effective jamming area in the azimuth direction. As seen from Figure 5a, for the nonlinear-trajectory airborne SAR with 5000 m flight height, 1 m resolution, and 0.5 m maximum trajectory offsets, the HDE algorithm can cover the deceptive jamming with a range width of about 7 km. The restriction of the HDE algorithm on the maximum trajectory deviation \(d_{max}\) is mainly referred to in Equations (40), (41), (43), (52) and (57).