## 4 Performance Evaluation
### Ergodic Sum Rate of the Spectrum Isolation Method  
Based on the assumptions given above, we study the ergodic sum rate of the conventional spectrum isolation transmission method. As the HS is much stronger than the UTs, the signal power satisfies \(P=\alpha P_{0}\), \(0<\alpha<1\). Define \(\beta=P_{0}/\sigma^{2}\) and \(\mathbb{E}[R]=\mathbb{E}[R_{0}]+\mathbb{E}[R_{1}]+\ldots+\mathbb{E}[R_{M}]\). The expression of ergodic sum rate is given in the following theorem. **Theorem 1**: _The ergodic sum rate of the spectrum isolation method for the SATCOM star network is given by_  
\[\mathbb{E}[R] = \mathbb{E}[R_{0}]+\sum_{i=1}^{M}\mathbb{E}[R_{i}] \tag{25}\] \[= \frac{\tau B}{\ln 2}\int_{0}^{\frac{|\alpha|^{2}\beta}{\tau B}} \frac{e^{-\frac{1}{L_{1}}\frac{\tau B\tau}{|\alpha|^{2}\beta-\tau B\tau}}}{1+x }dx+\frac{B-\tau B}{M}\frac{1}{\ln 2}\sum_{i=1}^{M}\int_{0}^{\frac{M(|x|^{2}\alpha \beta}{(B-\tau B)}}\frac{e^{-\frac{1}{L_{0}}\frac{(B-\tau B)\tau}{M(|x|^{2} \alpha\beta-(B-\tau B)\tau}}}{1+x}dx,\]  
_where \(L_{1}=\kappa_{1}G_{r}l^{2}(d)E_{t}\) and \(L_{0}=G_{r,0}l^{2}(d)E_{t}\)._  
According to (12) and (13), the ergodic sum rate can be expressed as  
\[\mathbb{E}[R] = \mathbb{E}[R_{0}]+\sum_{i=1}^{M}\mathbb{E}[R_{i}] \tag{26}\] \[= \tau B\mathbb{E}[\log_{2}(1+\frac{|\mathbf{v}_{1}^{\rm H}\mathbf{ H}_{1}\mathbf{w}_{ULS}\mathbf{v}_{0,5}^{\rm H}\mathbf{g}_{0}|^{2}P_{0}}{(| \mathbf{v}_{1}^{\rm H}\mathbf{H}_{1}\mathbf{w}_{ULS}|^{2}+1)\tau B\sigma^{2}} )]+\sum_{i=1}^{M}\frac{B-\tau B}{M}\mathbb{E}[\log_{2}(1+\frac{|\mathbf{h}_{0 }^{\rm H}\mathbf{w}_{0,5}\mathbf{v}_{ULS}^{\rm H}\mathbf{G}_{i}\mathbf{w}_{i}| ^{2}P}{(|\mathbf{h}_{0}^{\rm H}\mathbf{w}_{0,5}|^{2}+1)\frac{B-\tau B}{M} \sigma^{2}})].\]  
With Assumptions 1 and 2, the above equation can be further calculated as  
\[\mathbb{E}[R]=\tau B\mathbb{E}[\log_{2}(1+\frac{|h_{1}|^{2}|g_{0}|^{2}P_{0}}{ (|h_{1}|^{2}+1)\tau B\sigma^{2}})]+\sum_{i=1}^{M}\frac{B-\tau B}{M}\mathbb{E}[ \log_{2}(1+\frac{|h_{0}|^{2}|g_{i}|^{2}P}{(|h_{0}|^{2}+1)\frac{B-\tau B}{M} \sigma^{2}})]. \tag{27}\]  
Since \(\rho_{i}\sim\mathcal{CN}(0,1)\), \(i\in\{0,1,\ldots,M\}\), we have \(h_{1}\sim\mathcal{CN}(0,L_{1})\) and \(h_{0}\sim\mathcal{CN}(0,L_{0})\), where \(L_{1}=\kappa_{1}G_{r}l^{2}(d)E_{t}\) and \(L_{0}=G_{r,0}l^{2}(d)E_{t}\). Then, \(|h_{1}|^{2}\sim\exp(1/L_{1})\) and \(|h_{0}|^{2}\sim\exp(1/L_{0})\) can be obtained. According to the process adopted in [28], the ergodic rate of the HS signal can be written as  
\[\mathbb{E}[R_{0}]=\tau B\mathbb{E}[\log_{2}(1+\frac{|h_{1}|^{2}|g_{0}|^{2}P_{ 0}}{(|h_{1}|^{2}+1)\tau B\sigma^{2}})]=\frac{\tau B}{\ln 2}\int_{0}^{\infty} \frac{1-F_{X}(x)}{1+x}dx, \tag{28}\]  
where \(X=\frac{|h_{1}|^{2}|g_{0}|^{2}P_{0}}{(|h_{1}|^{2}+1)\tau B\sigma^{2}}\). As \(F_{X}(x)=\Pr(\frac{|h_{1}|^{2}|g_{0}|^{2}P_{0}}{(|h_{1}|^{2}+1)\tau B\sigma^{2 }}\leq x)\), we have  
\[F_{X}(x)=\Pr(\frac{|h_{1}|^{2}|g_{0}|^{2}P_{0}}{(|h_{1}|^{2}+1)\tau B\sigma^{2 }}\leq x)=\Pr(|h_{1}|^{2}(1-\frac{\tau B\sigma^{2}x}{|g_{0}|^{2}P_{0}})\leq \frac{\tau B\sigma^{2}x}{|g_{0}|^{2}P_{0}}). \tag{29}\]  
In case of \(1-\frac{\tau B\sigma^{2}x}{|g_{0}|^{2}P_{0}}>0\), i.e., \(x<\frac{|g_{0}|^{2}P_{0}}{\tau B\sigma^{2}}\), (29) can be calculated as  
\[F_{X}(x)=\Pr(|h_{1}|^{2}\leq\frac{\tau B\sigma^{2}x}{|g_{0}|^{2}P_{0}-\tau B \sigma^{2}x})=1-e^{-\frac{1}{L_{1}}\frac{\tau B\sigma^{2}x}{|g_{0}|^{2}P_{0}- \tau B\sigma^{2}x}} \tag{30}\]  
When \(|g_{0}|^{2}P_{0}-GPy-2\tau B\sigma^{2}y>0\), i.e., \(y<\frac{|g_{0}|^{2}P_{0}}{GP+2\tau B\sigma^{2}}\), the above equation becomes  
\[F_{Y}(y)=\Pr(|h_{1}|^{2}\leq\frac{\tau B\sigma^{2}y}{|g_{0}|^{2}P_{0}-GPy-2\tau B \sigma^{2}y})=1-e^{-\frac{1}{L_{1}}\frac{\tau B\sigma^{2}y}{|g_{0}|^{2}P_{0}- GPy-2\tau B\sigma^{2}y}} \tag{38}\]  
Otherwise, \(F_{Y}(y)=1\).