## 3 Interference Analysis and Dynamic Frequency Allocation
### Interference Analysis in the Worst Case  
When the LEO beam approaches the co-channel beam of the GEO satellite, the CCI becomes strong, and the signal quality of both systems declines accordingly. During the approach, there is a worst-case situation for both GEO and LEO users which means the signal quality is the lowest at every moment. Figure 3a depicts the geometrical configuration of the worst case for the GEO user. The solid line denotes the desired link and the black dashed denotes the interference link. When the LEO satellite is away from the GEO satellite, the worst case occurs if the GEO user locates at the edge of the GEO beam, where the desired link is the weakest, and the interference link is the strongest. Contrarily, when the LEO satellite has moved into the GEO beam, the worst case occurs if the GEO user is on the extension line between the GEO and LEO satellites. Because the quality of the desired link is almost changeless in the desired beam, the situation with the strongest interference link signifies the worst case. The two cases are shown in the left and right sides of Figure 3a, respectively. Basedon the geometrical analysis, the off-boresight angles and the distances related to the GEO user can be calculated as follows:  
\[\theta_{\mathrm{T,D}}=0.5\theta_{\mathrm{G,3 dB}}, \tag{5}\]  
\[d_{\mathrm{D}}=r\sin\big{(}\pi-\theta_{\mathrm{T,D}}-\arcsin\big{(}B\sin\theta_ {\mathrm{T,D}}/r\big{)}\big{)}\big{/}\sin\theta_{\mathrm{T,D}}, \tag{6}\]  
\[d_{\mathrm{I}}=\sqrt{A^{2}+r^{2}-2 Ar\cos\big{(}l/r-\pi+\theta_{\mathrm{T,D}}+ \arcsin\big{(}B\sin\theta_{\mathrm{T,D}}/r\big{)}\big{)}}, \tag{7}\]  
\[\theta_{\mathrm{T,I}}=\arcsin\big{(}r\sin\big{(}l/r-\pi+\theta_{\mathrm{T,D}}+ \arcsin\big{(}B\sin\theta_{\mathrm{T,D}}/r\big{)}\big{)}\big{/}d_{\mathrm{I}} \big{)}\,, \tag{8}\]  
\[\theta_{\mathrm{R,I}}=l/r+\theta_{\mathrm{T,I}}+\theta_{\mathrm{T,D}}, \tag{9}\]  
\[\theta_{\mathrm{R,D}}=0, \tag{10}\]  
when \(l\leq r\left(\pi-0.5\theta_{\mathrm{G,3 dB}}-\arcsin\big{(}B\sin\theta_{ \mathrm{T,D}}/r\big{)}\right)\),  
\[\theta_{\mathrm{T,D}}=\arcsin\bigg{(}A\sin\big{(}l/r\big{)}\bigg{/}\sqrt{A^{2 }+B^{2}-2 AB\cos\big{(}l/r\big{)}}\bigg{)}\,, \tag{11}\]  
\[\theta_{\mathrm{T,I}}=\arcsin\big{(}B\sin\theta_{\mathrm{T,D}}/A\big{)}\,, \tag{12}\]  
\[d_{\mathrm{I}}=r\sin\big{(}\pi-\theta_{\mathrm{T,I}}-\arcsin\big{(}B\sin\theta _{\mathrm{T,D}}/r\big{)}\big{)}\big{/}\sin\theta_{\mathrm{T,I}}, \tag{13}\]  
Figure 3: Geometrical configuration in the worst case. (**a**) GEO user; (**b**) LEO user. \[d_{\mathrm{D}}=r\sin\big{(}\pi-\theta_{\mathrm{T,D}}-\arcsin\big{(}B\sin\theta_ {\mathrm{T,D}}/r\big{)}\big{)}\big{/}\sin\theta_{\mathrm{T,D}}, \tag{6}\]0  
\[d_{\mathrm{D}}=r\sin\big{(}\pi-\theta_{\mathrm{T,D}}-\arcsin\big{(}B\sin\theta_ {\mathrm{T,D}}/r\big{)}\big{)}\big{/}\sin\theta_{\mathrm{T,D}}, \tag{6}\]1  
where T and R in the subscript denote the transmitter- and receiver-related variables, respectively; D and I denote the desired- and interference-related variables, respectively. \(A=r+h_{\rm L}\) and \(B=r+h_{\rm G}\), where \(h_{\rm L}\) and \(h_{\rm G}\) denote the orbital altitudes of the GEO and LEO satellites, respectively; \(r\) denotes the Earth radius; \(\theta_{\rm G,3 dB}\) denotes the angle corresponding to 3 dB beamwidth for GEO, and \(l\) is the distance between the boresight intersections of GEO and LEO beams. Similar to the GEO user, there are two typical geometrical configurations of the worst case for the LEO user, as shown in Figure 3b.