# High Reliable Uplink Transmission Methods in GEO-LEO Heterogeneous Satellite Network
## 4 Diversity Order Analysis  
To provide more insights regarding the outage probability with ordered channel, we provide the diversity order as an asymptotic reliability metric [28; 29]. In the context, the diversity order can be defined as  
\[D=-\lim_{\beta\rightarrow\infty}\frac{\log(Pr^{\infty}(\beta))}{\log\beta}, \tag{24}\]  
where \(Pr^{\infty}(\beta)\) denotes the asymptotic outage probability. Next, we analyze the asymptotic outage probability for \(U_{1}\) and \(U_{2}\).  
The asymptotic outage probability of \(U_{1}\) in high-SNR region is provided by \[\begin{array}{l}Pr_{1}^{\infty}(\beta)\\ =\frac{\chi\pi}{N}\sum_{n=1}^{N}\sqrt{1-x_{n}^{2}}\Big{(}\epsilon_{1,J}\sum_{k=0} ^{\infty}\frac{(m_{1,l})\kappa_{1,J}^{k}}{(k!)^{2}\kappa_{1,J}^{k+1}(k+1)}( \frac{T_{1}(y_{n,k}K_{2}\beta+1)}{K_{1}(1-a)\beta}\kappa_{1,J})^{k+1}-\epsilon _{1,J}\sum_{k=0}^{\infty}\frac{(m_{1,l})\kappa_{1,J}^{k}}{(k!)^{2}\kappa_{1,J}^ {k+1}}\frac{(y_{n,k}\kappa_{1,J})^{k+1}}{k+1}\Big{)}\\ \quad\times\epsilon_{2,J1}F_{1}(m_{2,l};1;v_{2,l}y_{n,1}).\end{array} \tag{25}\]  
Proof.: In (22), we have \(\gamma(k+1,\frac{T_{1}(y_{n,k}K_{2}\beta+1)}{K_{1}(1-a)\beta}\kappa_{1,J})= \sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{T_{1}(y_{n,k}K_{2}\beta+1)}{K_{1}(1-a )\beta}\kappa_{1,J})^{k+1}}{n!(k+1+n)}\). When \(\beta\rightarrow\infty\), \(y_{n,1}=\frac{(x_{n}+1)\chi}{2}\to 0\). Thus, \(\frac{T_{1}(y_{n,k}K_{2}\beta+1)}{K_{1}(1-a)\beta}\kappa_{1,J}\to 0\) still holds in high-SNR region. In this case, \(\gamma(k+1,\frac{T_{1}(y_{n,k}K_{2}\beta+1)}{K_{1}(1-a)\beta}\kappa_{1,J}) \approx\frac{\Big{(}\frac{T_{1}(y_{n,k}K_{2}\beta+1)}{K_{1}(1-a)\beta}\kappa_{ 1,J}\Big{)}^{k+1}}{k+1}\). Consequently, the asymptotic outage probability of \(U_{1}\) can be derived as (25).  
The corollary is proved.  
**Remark 1**.: _Substituting (25) into (24), we can derive the diversity order of the beam-center user as \(2\)._  
Following the similar procedure, we have \(S_{1}^{\infty}(\beta)=2\epsilon_{2,J}\sum_{k=0}^{\infty}\frac{(m_{2,l})\kappa_ {2,J}^{k}}{(k!)^{2}\kappa_{2,J}^{k+1}(k+1)}\Big{(}\frac{T_{2}}{\alpha\beta R_{ 2}^{2}}\kappa_{2,J}\Big{)}^{k+1}\)  
\(-\frac{T_{2}\pi}{a\beta R_{2}^{2}}\sum_{n=1}^{N}\sqrt{1-x_{n}^{2}}\epsilon_{2,J1}F_{1}(m_{2};1;v_{2,l}y_{n,2})\epsilon_{1,J}\sum_{k=0}^{\infty}\frac{(m_{ 1,l})\kappa_{1,J}^{k}}{(k!)^{2}\kappa_{1,J}^{k+1}(k+1)}\big{(}\zeta\kappa_{1, J}\big{)}^{k+1}\). Then, the asymptotic outage probability of \(U_{2}\) is derived in the following corollary.  
**Corollary 2**.: _The asymptotic outage probability of \(U_{2}\) in high-SNR region is provided by_  
\[\begin{array}{l}Pr_{2}^{\infty}(\beta)\\ =Pr_{1}^{\infty}(\beta)+S_{1}^{\infty}(\beta)\\ =\frac{\chi\pi}{N}\sum_{n=1}^{N}\sqrt{1-x_{n}^{2}}\bigg{(}\epsilon_{1,J}\sum_{ k=0}^{\infty}\frac{(m_{1,l})\kappa_{1,J}^{k}}{(k!)^{2}\kappa_{2,J}^{k+1}(k+1)}( \frac{T_{1}(y_{n,k}K_{2}\beta+1)}{K_{1}(1-a)\beta}\kappa_{1,J})^{k+1}-\epsilon _{1,J}\sum_{k=0}^{\infty}\frac{(m_{1,l})\kappa_{1,J}^{k}}{(k!)^{2}\kappa_{1,J}^ {k+1}}\frac{(y_{n,k}\kappa_{1,J})^{k+1}}{k+1}\bigg{)}\\ \quad\times\epsilon_{2,J1}F_{1}(m_{2,l};1;v_{2,l}y_{n,1})+2\epsilon_{2,J}\sum_ {k=0}^{\infty}\frac{(m_{2,l})\kappa_{2,J}^{k}}{(k!)^{2}\kappa_{2,J}^{k+1}(k+1) }\big{(}\frac{T_{2}}{\alpha\beta R_{2}}\kappa_{2,J}\Big{)}^{k+1}\\ -\frac{T_{2}\pi}{a\beta R_{2}^{2}N}\sum_{n=1}^{N}\sqrt{1-x_{n}^{2}}\epsilon_{2,J1}F_{1}(m_{2};1;v_{2,l}y_{n,2})\epsilon_{1,J}\sum_{k=0}^{\infty}\frac{(m_{1, l})\kappa_{1,J}^{k}}{(k!)^{2}\kappa_{1,J}^{k+1}(k+1)}\big{(}\zeta\kappa_{1, J}\big{)}^{k+1}.\end{array} \tag{26}\]  
**Remark 2**.: _Substituting (26) into (24), we can derive the diversity order of the beam-edge user as \(1\)._