# Resource Allocation for Cognitive LEO Satellite Systems: Facilitating IoT Communications
## 3 Asymptotic Analysis  
In this section, we analyze the asymptotic SINRs for both primary and secondary systems in order to allocate the resource for the two systems.  
We consider a large cognitive LEO SatCom system, in which the number of users is large, i.e., \(U\rightarrow\infty\) and \(K\rightarrow\infty\). To support a large number of users, it is reasonable toscale up \(N\) as well, i.e., \(N\rightarrow\infty\), but \(U/N\) converges to a constant parameter \(\alpha_{1}\), which represents the legacy satellite system load. Similarly, we have that \(K/N\) converges to a constant parameter \(\alpha_{2}\), which represents the IoT system load.  
To analyze the asymptotic SINR, we first present the following proposition.  
**Proposition 1** (Theorem 3.1 of [42]).: _For a symbol-synchronous multi-access spread-spectrum system with spreading gain \(N\), the SINR \(\gamma_{1}\) for the \(1\)-st user in a M user system is deterministic and approximately satisfies,_  
\[\gamma_{1}=\frac{p_{1}}{\sigma^{2}+\frac{1}{N}\sum_{i=2}^{M}I(p_{i},p_{1}, \gamma_{1})}, \tag{13}\]  
_where_  
\[I(p_{i},p_{1},\gamma_{1}){=}\frac{p_{i}p_{1}}{p_{1}+p_{i}\gamma_{1}}, \tag{14}\]  
_and \(p_{i}\) denotes the received power of the user \(i\)._  
From Proposition 1, the asymptotic SINR is determined by the received power for each user. Based on (13) and (14), the asymptotic SINR of the \(u\)-th legacy satellite user satisfies:  
\[\gamma_{\text{u,P}}=\frac{P_{\text{u}}|h_{\text{u}}|^{2}}{\frac{1}{N}\left( \sum_{i=1,i\
eq\frac{p_{i}|h_{\text{u}}|^{2}P_{\text{u}}|h_{\text{u}}|^{2}}{P_ {\text{u}}|h_{\text{u}}|^{2}}+\sum_{i=1}^{K}\frac{P_{\text{u}}|h_{\text{u}}|^ {2}P_{\text{u}}|f_{\text{u}}|^{2}}{P_{\text{u}}|h_{\text{u}}|^{2}+P_{\text{u }}|h_{\text{u}}|^{2}+P_{\text{u}}|h_{\text{u}}|^{2}\gamma_{\text{u,P}}}}\right) +\sigma^{2}}, \tag{15}\]  
As both the primary and secondary systems are based on CDMA, the method for analyzing the asymptotic SINR for the IoT users is the same as the one for the legacy satellite system. Thus, the asymptotic SINR of the \(k\)-th IoT user is given by:  
\[\gamma_{\text{ks}}=\frac{\bar{P}_{k}|f_{k}|^{2}}{\frac{1}{N}\left(\sum_{i=1,i \
eq\frac{p_{i}|f_{k}|^{2}|f_{k}|^{2}}{P_{\text{u}}|f_{k}|^{2}+P_{\text{u}}|f _{k}|^{2}}+\sum_{i=1}^{K}\frac{P_{\text{u}}|f_{k}|^{2}P_{\text{u}}|h_{\text{u} }|^{2}}{P_{\text{u}}|f_{k}|^{2}+P_{\text{u}}|f_{k}|^{2}\gamma_{\text{ks}}}} +\sigma^{2}\right.}, \tag{16}\]  
From (15) and (16), we find that the asymptotic SINRs for both primary and secondary systems are related to the received power for all of the links. To simplify the process of resource allocation, we assume that the received powers for legacy satellite users are the same one, which is given by \(q\). This is achieved by perfectly power control. Similarly, the received power for the IoT users is the same one, which is given by \(p\). Then, (15) can be simplified as  
\[\gamma_{\text{u,P}}=\frac{q}{\sigma^{2}{+}\frac{U}{N}\frac{q}{1+\gamma_{\text{ cap}}}+\frac{K}{N}\frac{pq}{p+qT_{\text{u,P}}}}, \tag{17}\]  
which gives  
\[\gamma_{p}=\frac{q}{\frac{q_{1}q}{\Gamma+\gamma_{p}}+\frac{q_{2}pq}{q+pT_{p}} +\sigma^{2}}. \tag{18}\]  
Similarly, (16) can be simplified as  
\[\gamma_{s}=\frac{p}{\frac{q_{2}p}{\Gamma+\gamma_{s}}+\frac{q_{1}pq}{p+q\gamma_ {s}}+\sigma^{2}}. \tag{19}\]