# 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 Power Affecting the Pixel after SAR Processing
#### 2.4.1 GEO-TO-LEO RFI  
When the beam coverage overlaps, RFI between GEO SARs and LEO SARs may occur. The beam coverage of GEO SARs is usually much larger than that of LEO SARs. Therefore, only LEO SAR beam coverage should be considered. As shown in Figure 3, the red frame represents the beam coverage of a LEO SAR (the antenna of LEO SARs is usually rectangular). Point A, B, C, and D are illuminated by both the LEO SAR and the GEO SAR at the same time. Thus, the RFI, which is transmitted by the GEO SAR and scattered by these points, would affect the LEO SAR. The RFI scattered by point A produces a yellow interference zone after the SAR processing. Since the GEO-to-LEO RFI scattered by point A and point B has the same parameter for (9), according to the principle of SAR, point B, located in the same range cell as point A, would produce the same interference zone as point A. The same is true for point C and D. Point O is at the junction of the yellow interference zone (generated by A and B) and the pink interference zone (generated by C and D). Based on the geometry, the interference zone formed by all the units in zone ABCD will affect cell O. Assuming there are N cells in total, the RFI power received by unit O can be written as \(\sum_{i=1}^{N}P_{i}\), where \(P_{i}(i=1,2,\ldots,N)\) is the RFI power of the \(i\)th cell. Figure 3: Schematic diagram of RFI; Point A, B, C, D and O are the five targets in the scene. A and B are in the same range cell, and the same is true for point C and D. Point O is the center of ABCD. The red frame represents the beam coverage of the LEO SAR and the yellow interference zone is from the RFI signal at point A and the pink interference zone is from the RFI signal at point D.  
Therefore, the size of zone ABCD is the key to calculate the RFI power affecting cell O. Obviously, the size of zone ABCD can be estimated by the product of AB and BC. AB is the azimuth direction length of beam coverage of LEO SAR on the ground. BC is the length of the interference zone in the range direction; we call it \(L_{1}^{G-L}\), which can be written as  
\[L_{1}^{G-L}=T_{p}c/(2\sin\theta_{LEO}) \tag{11}\]  
where \(\theta_{LEO}\) is the incidence angle of LEO SAR and \(T_{p}\) is the time width after pulse compression in the range direction; according to (9), it can be written as  
\[T_{p}=\frac{\min\left(k_{LEO}T_{pLEO},k_{GEO}T_{pGEO}\right)\cdot\left|k_{GEO} -k_{LEO}\right|}{k_{LEO}k_{GEO}} \tag{12}\]  
where \(k_{GEO}\) and \(T_{pGEO}\) are the FM rate and pulse width of GEO SAR, respectively. \(k_{LEO}\) and \(T_{pLEO}\) are the FM rate and pulse width of LEO SAR, respectively. Finally, the receiving aperture area can be expressed as  
\[A_{s}^{G-L}=A_{aLEO}R_{LEO}T_{p}c/(2\sin\theta_{LEO}) \tag{13}\]  
where \(A_{aLEO}\) and \(R_{LEO}\) are the beamwidth and slant range of LEO SAR, respectively. Generally, mismatch occurs for GEO-to-LEO RFI in both the range direction and azimuth direction. In this paper, we focus on the case that sampling rate satisfies the Nyquist theorem in range-direction. Then, according to (10), the gain of GEO-to-LEO RFI in the range direction can be written as  
\[G_{r}^{G-L}=f_{sLEO}^{2}/|k_{GEO}-k_{LEO}| \tag{14}\]  
where \(f_{sLEO}\) is the sampling rate of the LEO SAR. When the Nyquist theorem is not satisfied, the gain can be given by a numerical simulation, which is detailed below, but the range direction spectral aliasing is not discussed in our paper. The PRF of LEO SARs is usually much higher than that of GEO SARs; that is, a LEO SAR receives multiple pulses from itself before receiving a pulse from a GEO SAR (as shown in Figure 2), which is equivalent to down-sampling, causing spectrum aliasing in the azimuth direction. Therefore, the gain of GEO-to-LEO RFI in the azimuth direction is given by a numerical simulation. The step is shown in Table 1, where the subscript SAR represents the disturbed SAR, and the subscript \(i\) represents the RFI source.