## 4 Performance Analysis
### Ergodic Rate  
#### 4.1.1 Co-Carrier-Based Direct Combination  
Under Assumption 1, we have \(|g_{k,m}|=|\rho_{k,m}||l(d_{k,m})\sqrt{E_{S}(\eta_{k,m})}|=|\rho_{k,m}||\beta_ {k,m}|\), where \(g_{k,m}\sim\mathcal{CN}(0,|\beta_{k,m}|^{2})\). Similarly, we can derive \(g_{m,m}\sim\mathcal{CN}(0,|\beta_{m,m}|^{2})\) for the beam-center user under satellite \(m\). In this case, SINR of the \(k\)-th UT in (4) can be further computed as  
\[SINR_{k}=\frac{|\sum_{m=1}^{M}g_{k,m}|^{2}P_{k}}{\sum_{m=1}^{M}|g_{k,m}|^{2}P+ \sigma^{2}}. \tag{10}\]  
Further, the SNR of the beam-center UT can be calculated as  
\[SNR_{m,m}=\frac{|g_{m,m}|^{2}P}{\sigma^{2}}. \tag{11}\]  
Then, the achievable sum-rate of UT \(k\) can be expressed as  
\[R_{k}=\log_{2}(1+SINR_{k})=\log_{2}(1+\frac{|\sum_{m=1}^{M}g_{k,m}|^{2}P_{k}}{ \sum_{m=1}^{M}|g_{k,m}|^{2}P+\sigma^{2}}). \tag{12}\]  
Accordingly, we have  
\[R_{m}=\log_{2}(1+SINR_{m,m})=\log_{2}(1+\frac{|g_{m,m}|^{2}P}{\sigma^{2}}). \tag{13}\]  
Since UT \(k\) could receive \(M\) signal paths from the \(M\) satellites, and each satellite could serve one beam-center UT simultaneously by applying the co-carrier-based DC method with one resource block. The ergodic sum-rate of the co-carrier-based DC method can be expressed as \(\mathbb{E}[R_{DC}]=\mathbb{E}[R_{k}]+\sum_{m=1}^{M}\mathbb{E}[R_{m}]\). Let \(\delta=P_{k}/\sigma^{2}\), \(|\beta_{k,1}|^{2}=\ldots=|\beta_{k,M}|^{2}=|\beta_{k}|^{2}\), and \(|\beta_{1,1}|^{2}=\ldots=|\beta_{M,M}|^{2}=|\beta_{m}|^{2}\). Then, we can derive the following theorem. **Theorem 1**. : _The ergodic sum-rate of UIT \(k\) and the co-carrier-based DC method in multiple nodes LEO SATCOM can be given as_  
\[\mathbb{E}[R_{DC}] =\mathbb{E}[R_{k}]+\sum_{m=1}^{M}\mathbb{E}[R_{m}]\] \[=\frac{\delta M|\beta_{k}|^{2-2M}}{\ln 2}\sum_{i=1}^{n}\omega_{i}B_{ k}(\delta M|\beta_{k}|^{2}x_{i})-\frac{Me^{1/(|\beta_{m}|^{2}\alpha\delta)}}{\ln 2 }\mathrm{E}(-\frac{1}{|\beta_{m}|^{2}\alpha\delta}), \tag{14}\]  
_where \(B_{k}(x)=\frac{(xa/(M|\beta_{k}|^{2})+\frac{1}{|\beta_{k}|^{2}})^{-M}}{1+x}\), \(x_{i}\) and \(\omega_{i}\) denote the Gauss-Laguerre quadrature nodes and weights over \([0,+\infty]\)[22]. To be specific, \(\mathrm{E}(x)=-\int_{-x}^{\infty}\frac{e^{-r}}{r}dr\) is the exponential integral function [23]._  
Proof. : For notational convenience, we rewrite the ergodic rate for UT \(k\) as  
\[R_{k} =\log_{2}(1+\underbrace{\overbrace{\sum_{m=1}^{M}g_{k,m}|^{2}P_{ k}}^{\sum_{m=1}^{M}g_{k,m}}^{\sum_{m=1}^{M}g_{k,m}}^{\sum_{m=1}^{M}|g_{k,m}|^{2}P_{ +\sigma^{2}}}}_{\sum_{m=1}^{M}|g_{k,m}|^{2}\sum_{p+\sigma^{2}}}).\]  
We have \(Y=|\sum_{m=1}^{M}g_{k,m}|^{2}\sim\exp(1/(M|\beta_{k}|^{2}))\) whose probability density function (PDF) is  
\[f_{Y}(y)=\frac{1}{M|\beta_{k}|^{2}}e^{-\frac{y}{M|\beta_{k}|^{2}}}. \tag{15}\]  
Since \(g_{k,m}\)s are independent and identically distributed, the PDF of \(Z\) can be calculated as  
\[f_{Z}(z)=\frac{(z^{M-1}e^{-1/|\beta_{k}|^{2}z})(1/|\beta_{k}|^{2})^{M}}{(M-1)!}. \tag{16}\]  
Then, \(\mathbb{E}[R_{k}]\) is given by  
\[\mathbb{E}[R_{k}]=\mathbb{E}[\log_{2}(1+\frac{YP_{k}}{ZP+\sigma^{2}})]=\frac {1}{\ln 2}\int_{0}^{\infty}\frac{1-F_{X}(x)}{1+x}dx, \tag{17}\]  
where \(X=\frac{YP_{k}}{ZP+\sigma^{2}}\) and \(F_{X}(x)\) is the cumulative distribution function (CDF). According to the PDF of \(Y\) and \(Z\), \(F_{X}(x)\) can be further computed as  
\[SNR_{m,m}=\frac{|g_{m,m}|^{2}P}{\sigma^{2}}. \tag{11}\]0 \[SNR_{m,m}=\frac{|g_{m,m}|^{2}P}{\sigma^{2}}. \tag{11}\]1 \[SNR_{m,m}=\frac{|g_{m,m}|^{2}P}{\sigma^{2}}. \tag{11}\]2 \[SNR_{m,m}=\frac{|g_{m,m}|^{2}P}{\sigma^{2}}. \tag{11}\]3 \[SNR_{m,m}=\frac{|g_{m,m}|^{2}P}{\sigma^{2}}. \tag{11}\]4 \[SNR_{m,m}=\frac{|g_{m,m}|^{2}P}{\sigma^{2}}. \tag{11}\]5According to ([23], 3.351.3), (18) can be further calculated as  
\[SNR_{m,m}=\frac{|g_{m,m}|^{2}P}{\sigma^{2}}. \tag{11}\]6  
Then, (17) can be derived as  
\[SNR_{m,m}=\frac{|g_{m,m}|^{2}P}{\sigma^{2}}. \tag{11}\]7  
Let \(B_{k}(x)=\frac{(\frac{x\alpha}{M|\beta_{k}|^{2}}+\frac{1}{|\beta_{k}|^{2}})^{ -M}}{1+x}\).