# Deceptive Jamming Detection for SAR Based on Cross-Track Interferometry
## 2 Principle of Deceptive Jamming Detection
### Interferometric Phase Analysis  
In the interferometric process, an envelope shift between the master image and the slave image is compensated by high precision image co-registration. After co-registration, the interferometric phase of Equations (10) and (11) is derived as [23]  
\[\Delta\phi_{real}=-\frac{2\pi}{\lambda}(R_{M}-R_{S}) \tag{12}\]  
Assuming that the slave image in Equation (6) is adjusted in the light of the master image, \(J_{S\_out}(t_{r},t_{a})\) after co-registration is rearranged as  
\[\begin{array}{ll}J_{S\_out}(t_{r},t_{a})\approx&GA\sigma(x,y)\sin c\Big{(}t _{r}-\frac{2R_{M}}{c}\Big{)}\sin c\big{(}t_{a}-t_{0}\big{)}\\ &\cdot\exp\Bigl{(}-j\frac{4\pi}{\lambda}R_{M}\Bigr{)}\exp\bigl{(}j\frac{2\pi}{ \lambda}\big{(}R_{MJ0}-R_{SJ0}\big{)}\bigr{)}\end{array} \tag{13}\]  
Then, the interferometric phase of Equations (5) and (13) is given by complex conjugate multiplication as [20]  
\[\Delta\phi_{deception}=-\frac{2\pi}{\lambda}\big{(}R_{MJ0}-R_{SJ0}\big{)}\approx -\frac{2\pi}{\lambda}\big{(}R_{MJ}-R_{SJ}\big{)} \tag{14}\]  
where \(R_{MJ}\) is the minimum slant range between the master antenna and the jammer and \(R_{SJ}\) is the minimum slant range between the slave antenna and the jammer.  
Equation (14) indicates that the phase difference of false target depends heavily on the relative position between the jammer and the two antennas, while not related to the terrain of imaging scene. In practice, the jammer position will be unchanged when SAR deceptive jamming occurs, which results in a phase constant for false targets on the interferogram. Namely, once the location of jammer is determined, the phase constant will not vary even if the jammer changes signal type or alters different deceptive regions on SAR image. Furthermore, Equation (12) shows that the surrounding real areas appear as varying InSAR phases due to different surface elevation. Therefore, the false phase is significantly different from the surrounding real area in the interferogram.  
As an example, a series of point targets along the range dimension have been simulated utilizing the system parameters listed in Table 1. After SAR imaging and co-registration of two SAR images, 700 pixels of interferometric phase are generated, as shown in Figure 2a. To jam 331 to 570 pixels and 611 to 660 pixels of the SAR image, we pre-designed the false elevation and repeated the SAR imaging and interferometry process. Then, the interferometric phase after jamming is obtained as shown in Figure 2b. Note that the jammer \(J\) is placed on \((0,179272.327,0)\) in the simulation. Additionally, when the minimum distance between the master antenna and the jammer is achieved, the instantaneous locations of the master antenna and the slave one are \((0,0,514800)\) and \((0,200,514800)\), respectively. According to Equation (14), the theoretical false phase can be derived with a value of 1.8891 rad (the red dotted line in Figure 2b). By comparing Figure 2a,b, the false phases significantly differ from the real ones and fluctuate around a constant value as expected, which provides an opportunity to exact the false part from the deceived SAR image.