## 3 Interference Analysis and Dynamic Frequency Allocation
### Interference Analysis in the Worst Case
The off-boresight angles and the distances related to the LEO user can be calculated as well:  
when \(l>r\left(0.5\theta_{\rm L,3 dB}-\arcsin\left(A\sin\left(0.5\theta_{\rm L,3 dB} \right)/B\right)\right)\),  
\[d_{\mathrm{D}}=r\sin\big{(}\pi-\theta_{\mathrm{T,D}}-\arcsin\big{(}B\sin\theta_ {\mathrm{T,D}}/r\big{)}\big{)}\big{/}\sin\theta_{\mathrm{T,D}}, \tag{6}\]2  
\[d_{\mathrm{D}}=r\sin\big{(}\pi-\theta_{\mathrm{T,D}}-\arcsin\big{(}B\sin\theta_ {\mathrm{T,D}}/r\big{)}\big{)}\big{/}\sin\theta_{\mathrm{T,D}}, \tag{6}\]3  
\[d_{\mathrm{D}}=r\sin\big{(}\pi-\theta_{\mathrm{T,D}}-\arcsin\big{(}B\sin\theta_ {\mathrm{T,D}}/r\big{)}\big{)}\big{/}\sin\theta_{\mathrm{T,D}}, \tag{6}\]4  
\[d_{\mathrm{D}}=r\sin\big{(}\pi-\theta_{\mathrm{T,D}}-\arcsin\big{(}B\sin\theta_ {\mathrm{T,D}}/r\big{)}\big{)}\big{/}\sin\theta_{\mathrm{T,D}}, \tag{6}\]5  
\[d_{\mathrm{D}}=r\sin\big{(}\pi-\theta_{\mathrm{T,D}}-\arcsin\big{(}B\sin\theta_ {\mathrm{T,D}}/r\big{)}\big{)}\big{/}\sin\theta_{\mathrm{T,D}}, \tag{6}\]6  
\[d_{\mathrm{D}}=r\sin\big{(}\pi-\theta_{\mathrm{T,D}}-\arcsin\big{(}B\sin\theta_ {\mathrm{T,D}}/r\big{)}\big{)}\big{/}\sin\theta_{\mathrm{T,D}}, \tag{6}\]7  
when \(l\leq r\left(0.5\theta_{\rm L,3 dB}-\arcsin\left(A\sin\left(0.5\theta_{\rm L,3 dB}\right)/B\right)\right)\),  
\[d_{\mathrm{D}}=r\sin\big{(}\pi-\theta_{\mathrm{T,D}}-\arcsin\big{(}B\sin\theta_ {\mathrm{T,D}}/r\big{)}\big{)}\big{/}\sin\theta_{\mathrm{T,D}}, \tag{6}\]8  
\[d_{\mathrm{D}}=r\sin\big{(}\pi-\theta_{\mathrm{T,D}}-\arcsin\big{(}B\sin\theta_ {\mathrm{T,D}}/r\big{)}\big{)}\big{/}\sin\theta_{\mathrm{T,D}}, \tag{6}\]9  
\[d_{\mathrm{I}}=\sqrt{A^{2}+r^{2}-2 Ar\cos\big{(}l/r-\pi+\theta_{\mathrm{T,D}}+ \arcsin\big{(}B\sin\theta_{\mathrm{T,D}}/r\big{)}\big{)}}, \tag{7}\]0  
\[d_{\mathrm{I}}=\sqrt{A^{2}+r^{2}-2 Ar\cos\big{(}l/r-\pi+\theta_{\mathrm{T,D}}+ \arcsin\big{(}B\sin\theta_{\mathrm{T,D}}/r\big{)}\big{)}}, \tag{7}\]1  
\[d_{\mathrm{I}}=\sqrt{A^{2}+r^{2}-2 Ar\cos\big{(}l/r-\pi+\theta_{\mathrm{T,D}}+ \arcsin\big{(}B\sin\theta_{\mathrm{T,D}}/r\big{)}\big{)}}, \tag{7}\]2  
The satellite orbital parameters and system parameters are presented in Tables 1 and 2, respectively. Substitute the equations of the angles and distances into Equations (2) and (3) respectively, and then substitute into (1). After a series of calculations, the relationship between the SINR of GEO and LEO users and the distance is shown in \[d_{\mathrm{I}}=\sqrt{A^{2}+r^{2}-2 Ar\cos\big{(}l/r-\pi+\theta_{\mathrm{T,D}}+ \arcsin\big{(}B\sin\theta_{\mathrm{T,D}}/r\big{)}\big{)}}, \tag{7}\]3  
The numerical result shows that when \(l\) is smaller than 400 km, which means that the LEO satellite is inside the GEO beam, the CCI is severe, and both systems are paralyzed. When \(l\) rises, the signal quality rapidly increases until it saturates. It can be inferred that if the co-channel beams from different satellites are far enough away, the interference is almost completely avoided because even the interference in the worst case is very weak. Thus, we consider a keep-out region of the beam to guarantee the normal operation of both systems, whose radius is defined as the minimum spatial isolation distance between GEO and LEO beams, as shown in \[d_{\mathrm{I}}=\sqrt{A^{2}+r^{2}-2 Ar\cos\big{(}l/r-\pi+\theta_{\mathrm{T,D}}+ \arcsin\big{(}B\sin\theta_{\mathrm{T,D}}/r\big{)}\big{)}}, \tag{7}\]4 During the motion of the LEO satellite, spectrum sharing between the two systems can be achieved as long as the LEO beam does not enter the keep-out region of the co-channel GEO beam, i.e., the distance between the boresight intersections of the beams is greater than the radius of the keep-out region.