## 1 Introduction  
Synthetic Aperture Radar (SAR) is an active microwave imaging system that uses electromagnetic waves for two-dimensional high-resolution imaging, which has many advantages, such as all-day, all-weather, strong penetration capability, high processing gain, and strong anti-jamming capability [1]. It is widely used in strategic reconnaissance, strike effect assessment, and other military occasions and has become a significant method to obtain information in modern warfare. Its powerful and flexible functions can substantially threaten high-value strategic targets, military facilities, and intelligence security. Therefore, to protect sensitive targets and sensitive areas, electronic countermeasures against SAR have received extensive attention and development [2; 3; 4; 5]. According to the different jamming effects, active jamming techniques for SAR are divided into barrage jamming and deceptive jamming. Barrage jamming uses high-powered noise to attenuate the signal to jamming ratio within a specific region to cover up the real target [6; 7; 8; 9].The deceptive jamming confuses target identification without causing enemy awareness by directly generating or modulating-retransmitting false target echo signals and implanting false electromagnetic features in SAR radar images [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24]. Comparedwith barrage jamming, deceptive jamming is characterized by higher concealment, lower power consumption and more flexible application scenarios. Therefore, deceptive jamming is more attractive and promising. The modulation and retransmission mechanism is currently the mainstream research approach for SAR deception jamming [25; 26]. The deceptive jammer based on the modulation-retransmission mechanism can be modeled as a linear time-invariant system. In each pulse repetition interval (PRI), the jamming system generates a frequency response function based on a series of parameters such as platform parameters, antenna parameters, and signal parameters of the SAR to be jammed [10; 11]. The basic scheme of modulation-retransmission deceptive jamming is to modulate and retransmit the spectrum of intercepted pulses with the frequency response function of the jamming system to produce a jamming signal, which is imaged by the SAR receiver to form a false target. The traditional algorithm is the point-by-point superposition straightforward calculation algorithm (SA), which directly calculates the difference in signal propagation delay between each scatter point in the deceptive jammer template and the jammer during each PRI. However, this algorithm involves the iterative operation by the double integral in continuous pulses, and it is challenging to guarantee real-time modulation calculation [11]. Several algorithms have been proposed to reduce the computational burden to fulfill real-time jamming and have been divided into two categories: azimuth time-domain processing and azimuth frequency-domain processing. The former reduces computational complexity through slant-range approximation equations and range-azimuth split-dimensional modulation, which can cause a loss of image quality to a certain extent, including the segmented modulation algorithms [14; 15], the multi-receiver approximation algorithms [16; 17] and the time-delay frequency-shifting algorithms with template segmentation [22]. The latter requires azimuth Fourier transform modulation processing on the template, which is relatively precise but faces obstacles in practicality due to some defects, such as the low-efficiency interpolation calculation [27; 28], including the inverse range-Doppler algorithm [12], the frequency domain three-stage algorithm [19], the inverse Omega-K algorithm [20], and the frequency domain pre-modulation algorithm [18]. All the above algorithms assume SAR has a nominal linear trajectory and perform partial calculations of slant range before real-time modulation.