# 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
#### 2.4.2 LEO-TO-GEO RFI  
According to Figure 3, \(A_{s}\) is related to the size of the interference zone. Like GEO-to-LEO RFI, the azimuth length of \(A_{s}\) corresponds to the azimuth direction length of the LEO SAR beam coverage on the ground. However, the range direction length of the interference zone generated by the LEO-to-GEO RFI is different from that of the GEO-to-LEO RFI (written as (11)). As shown in Figure 4, the LEO-to-GEO RFI can have a much larger range migration than GEO SAR signals due to the long integration time of the GEO SAR and the different velocity between the GEO SAR and the LEO SAR. However, there is little difference on the migration between the LEO SAR and the GEO-to-LEO RFI due to the short integration time of the LEO SAR. When the Back Projection (BP) algorithm is adopted, the RFI will be accumulated along the migration direction of the SAR signals. Thus, the interference zone generated by the LEO-to-GEO RFI will spread in the range direction due to its inconsistent migration direction with the GEO SAR signals, while the interference zone generated by the GEO-to-LEO RFI will remain unchanged due to its similar migration direction with the LEO SAR signals. As a result, the width \(L_{2}^{L-G}\) in the range direction is related to the slant range, which can be expressed as  
\[L_{2}^{L-G}=2(R_{\text{max}}-R_{\text{min}}) \tag{17}\]  
where \(R_{\text{max}}\) and \(R_{\text{min}}\) are the maximum and minimum slant range of LEO-to-GEO RFI. In summary, \(A_{s}\) of LEO-to-GEO RFI can be expressed as  
\[A_{s}^{L-G}=2(R_{\text{max}}-R_{\text{min}})A_{aLEO}R_{LEO} \tag{18}\]  
Similarly, we only discuss the case that the Nyquist theorem is satisfied in the range-direction. Like (14), the gain obtained by LEO-to-GEO RFI from range compression can be written as  
\[G_{r}^{L-G}=f_{sGEO}^{2}/|k_{GEO}-k_{LEO}| \tag{19}\]  
where \(f_{sGEO}\) is the sampling rate of the GEO SAR. Since the PRF of LEO SARs is much higher than that of GEO SARs, spectrum aliasing will occur in the azimuth-direction. Therefore, the gain \(G_{a}^{L-G}\) obtained by LEO-to-GEO  
Figure 4: Relationship between migration and the accumulation direction. \(L_{1}\) is the range direction length of the interference band before the BP algorithms (after the range compression). \(L_{2}\) is the range direction length of the interference band after the BP algorithms. (**a**) LEO-to-GEO RFI (**b**) GEO-to-LEO RFI. RFI in the azimuth-direction is also given by the numerical simulation shown in Table 1. In this case, \(k_{sSAR}\) is the FM rate of the GEO SAR in the azimuth direction, \(T_{sSAR}\) is the integration time of the GEO SAR, and \(k_{ai}\) is the FM rate of the LEO-to-GEO RFI in the azimuth direction. Furthermore, Step 3 should be skipped. However, based on Figure 4, the interference zone can spread in the range direction after the BP algorithms, and, as a result, the gain obtained by LEO-to-GEO RFI in the azimuth-direction can decrease; the gain after decreasing is referred to as \(\widetilde{G}_{a}^{L-G}\). Obviously, \(\widetilde{G}_{a}^{L-G}<G_{a}^{L-G}\). Referring to (11), the length of interference zone in the range direction before widening can be expressed as  
\[L_{1}^{L-G}=T_{p}c/2\sin\theta_{GEO} \tag{20}\]  
where \(\theta_{GEO}\) is the incidence angle of GEO SAR. Thus, the range direction length of the interference zone changes from \(L_{1}^{L-G}\) to \(L_{2}^{L-G}\), where \(L_{1}^{L-G}<L_{2}^{L-G}\). \(P_{1}\) represents the total power before widening, then it can be expressed as  
\[P_{1}=P_{\text{enter}}G_{r}^{L-G}G_{a}^{L-G}L_{1}^{L-G}A_{aLEO}R_{LEO} \tag{21}\]  
where \(P_{\text{enter}}\) is the RFI power entering the receiver. \(P_{2}\) denotes the total power after widening and can be written as  
\[P_{2}=P_{\text{enter}}G_{r}^{L-G}\widetilde{G}_{a}^{L-G}L_{2}^{L-G}A_{aLEO}R_ {LEO} \tag{22}\]  
Based on the conservation of energy, \(P_{1}=P_{2}\). It is apparent that \(\widetilde{G}_{a}^{L-G}=G_{a}^{L-G}L_{1}^{L-G}/L_{2}^{L-G}\).