# Modeling and Analysis of RFI Impacts on Imaging between Geosynchronous SAR and Low Earth Orbit SAR
## 2 Accurate Modeling of SINR in the Presence of RFI  
### RFI Impact Quantitative Analysis Model  
A sketch map of when a GEO SAR and a LEO SAR observe the same target is shown in Figure 1. O represents the position of the target, illuminated by a LEO SAR and a GEO SAR at the same time, L is the position of the LEO SAR, OL is the slant range of the LEO SAR, G is the position of the GEO SAR, OG is the slant range of the GEO SAR, \(\hat{x}\) is the unit vector of the projection of \(\overset{\rightarrow}{\text{OL}}\) on the scene plane, \(\hat{z}\) is the normal unit vector of the scene, \(\hat{y}\) is determined by the cross product of orthonormal basis \(\hat{z}\) and \(\hat{x}\), \(\theta_{1}\) is the incidence angle of the LEO SAR (or GEO-to-LEO RFI bistatic scattering angle to LEO SAR) and \(\theta_{2}\) is the incidence angle of the GEO SAR (or LEO-to-GEO RFI bistatic scattering angle to GEO SAR); \(\beta\) is the bistatic angle and \(\varphi\) is the out-of-plane angle of the bistatic scattering direction. The echo of a single target with RFI and noise can be expressed as [35]  
\[x=s+i+n \tag{1}\]  
where \(s\) is the target signal, \(i\) is the RFI signal, \(n\) is the noise. Then, SAR processing can be described as  
\[I_{m}=x\otimes G=s_{G}+i_{G}+n_{G} \tag{2}\]  
where \(I_{m}\) is the imaging result, \(G\) is the point target response of SAR processing, and \(s_{G}\), \(i_{G}\), \(n_{G}\) are the SAR processing result of \(s\), \(i\) and \(n\), respectively. Obviously, RFI enters the receiver and signal processing system with the target signal at the same time. As a result, the output image \(I_{m}\) is the linear summation in the complex domain, performing as noise or ghost targets and deteriorating the image. We adopted SINR to judge whether the RFI can be neglected or not. When the image SINR falls below 5 dB, the image quality is poor, leading to loss of detailed information [36]. Considering the SAR receiver, the definition of SINR is [32; 33]  
\[SINR=\frac{P_{s}}{P_{th}+P_{RFI}} \tag{3}\]  
where \(P_{s}\) is the power of \(s_{G}\), which can be obtained from the SAR radar equation [37].In digital signal processing, echo focusing is considered as coherent accumulation, thus, its amplitude gain can be written as \(N\)a \(\times\)\(N\)r, where \(N\)r and \(N\)a are the sampling point number in the range direction and the azimuth direction, respectively. Since, \(N\)r = \(T_{ps}\)\(f_{s}\) and \(N\)r = \(T_{s}\)\(PRF_{s}\), where \(T_{ps}\), \(f_{s}\), \(T_{s}\) and \(PRF_{s}\) are pulse width, sampling rate, integration time and pulse repetition frequency (PRF) of the disturbed SAR system, respectively, \(P_{s}\) can be written as  
\[P_{s}=P_{\text{s}}{G_{\text{t}}}^{2}\sigma_{0}\rho_{a}\rho_{gr}{\lambda_{s}}^ {2}(T_{ps}f_{s})^{2}(T_{s}\cdot PRF_{s})^{2}/(4\pi)^{3}{R_{s}}^{4} \tag{4}\]  
Figure 1: Schematic diagram when a GEO SAR and a LEO SAR observe the same target. where \(P_{ts}\), \(G_{ts}\), \(R_{s}\) and \(\lambda_{s}\) are the peak transmitted power, antenna gain, slant range and wavelength of the disturbed SAR system, respectively. \(\sigma_{0}\) is the normalized backscatter coefficient, \(\rho_{a}\) and \(\rho_{gt}\) are azimuth resolution and ground range resolution, respectively. \(P_{th}\) is the power of \(n_{G}\). Since the thermal noise is usually considered as a stochastic process, it is superimposed incoherently in SAR processing. As a result, the power gain of the thermal noise is \(Na\times Nr\). Then, \(P_{th}\) can be written as  
\[P_{th}=G_{th}k_{B}T_{th}B=\big{(}T_{ps}f_{s}\big{)}\big{(}T_{s}\cdot PRF_{s} \big{)}k_{B}T_{th}B \tag{5}\]  
where \(k_{B}=1.38\times 10^{-23}J/k\) is the Boltzmann constant, \(T_{th}\) is the thermal noise temperature, \(B\) is the bandwidth of the receiver and \(G_{th}\) is the gain caused by the incoherent accumulation of thermal noise. \(P_{RFI}\) is the power of \(i_{G}\).