# Performance and Requirements of GEO SAR Systems in the Presence of Radio Frequency Interferences
## 3 Bistatic Scattering RFI Impact on GEO SAR
### Doppler Filtering  
Specially, if our interesting target is point-like and isolated, the received pulses will have deterministic phases. In this case, Doppler filtering on RFI signal will have a good performance to weaken the RFI impact. Figure 6a shows the geometry difference between GEO SAR and LEO SAR. \(\mathbf{v_{s}}\) and \(\mathbf{v_{g}}\) are the velocity vectors of the LEO SAR system and the GEO SAR system, respectively. Vectors \(\mathbf{R_{s}}\) and \(\mathbf{R}\) represent the incidence direction (line of sight (LOS)) and the bistatic scattering direction, respectively. \(\vartheta\) is the LOS angle between \(\mathbf{v_{s}}\) and \(\mathbf{R_{s}}\). GEO SAR has a larger beam footprint (e.g., typically 400 km) than that of LEO SAR (e.g., typically 10 km in along-track). Moreover, the velocity of GEO SAR is small compared with that of a LEO SAR. Resultantly, the Doppler rate of GEO SAR is very small compared with that of a LEO SAR. To achieve the same azimuth resolution, the Doppler bandwidth of GEO SAR is much smaller (only dozens of Hz at most) than the Doppler bandwidth of a LEO SAR (hundreds to thousands Hz), which is shown in Figure 6b. Moreover, as the pulse repetition frequency (PRF) of GEO SAR is still smaller than the Doppler bandwidth of a LEO SAR, the Doppler frequency of a LEO SAR will be aliased. When doing the Doppler filtering in the imaging processing, a GEO system achieves the azimuth focusing by exploiting the small Doppler bandwidth. Therefore, the components of the LEO SAR RFI Doppler spectrum out of the Doppler bandwidth of GEO SAR could be filtered, as shown as the yellow regions in Figure 6b. It should be noticed that the aliased Doppler frequencies of the LEO SAR within the GEO SAR Doppler bandwidth cannot be filtered out. As the integration time of GEO SAR (hundreds of second or even hours) is far longer than that in a LEO SAR (typically 1 s), if we assume the LEO SAR observes the target within the integration time of the GEO SAR in a worst case (like a staring mode), the LEO SAR signal can be simplified as weighted by the envelope of GEO SAR. Moreover, the worst geometry relationship between the GEO SAR and  
Figure 6: (a) Geometry relationship between LEO SAR and GEO SAR; (b) Doppler bandwidths of GEO SAR and LEO SAR. LEO SAR is assumed as well, which makes the RFI signal of LEO SAR be weighted by the GEO SAR antenna pattern. Under the assumption, the received RFI signal from the LEO SAR is expressed as  
\[ \begin{split} s_{l}(t)&=\,A\sin\mathrm{c}^{2}\bigg{(} \frac{v_{g}L_{g}}{\lambda R_{0}}t\bigg{)}\exp\bigg{(}j\frac{4\pi R_{s}(t)}{ \lambda}\bigg{)}\\ &=\,A\sin\mathrm{c}^{2}\bigg{(}\frac{v_{g}L_{g}}{\lambda R_{0}}t \bigg{)}\exp\bigg{[}j\pi\bigg{(}\frac{4R_{0,I}}{\lambda}+f_{dc,I}t+f_{dr,I}t^{ 2}+\cdots\bigg{)}\bigg{]},\end{split}  \tag{17}\]  
where \(A\) (relates to SINR) is the amplitude of the RFI, \(L_{g}\) is the antenna size of GEO SAR, \(v_{g}\) is the velocity of GEO SAR, \(R_{0,I}\) is the shortest slant range of \(R_{s}\), \(f_{dr,I}\) is the Doppler rate of the LEO SAR, \(f_{dc,I}\) is the Doppler centroid frequency of the LEO SAR. Sinc function represents the envelope of the GEO SAR signal. The received GEO SAR signal is  
\[ \begin{split} s_{l}(t)&=\sin\mathrm{c}^{4}\bigg{(} \frac{v_{g}L_{g}}{\lambda R_{0}}t\bigg{)}\exp\bigg{(}j\frac{4\pi R(t)}{\lambda }\bigg{)}\\ &=\sin\mathrm{c}^{4}\bigg{(}\frac{v_{g}L_{g}}{\lambda R_{0}}t \bigg{)}\exp\bigg{[}j\pi\bigg{(}\frac{4R_{0}}{\lambda}+f_{dc,g}t+f_{dr,g}t^{ 2}+\cdots\bigg{)}\bigg{]},\end{split}  \tag{18}\]  
where \(R_{0}\) is the shortest slant range of GEO SAR, \(f_{dr,g}\) is the Doppler rate of the GEO SAR, \(f_{dc,g}\) is the Doppler centroid frequency of the GEO SAR. The energy of the LEO SAR RFI signal \(E\) after Doppler filtering is  
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