# Performance Analysis of a Cognitive RIS-NOMA in Wireless Sensor Network
## 3 Performance Analysis  
### The Channel Model  
In this section, we calculate new channel statistics for the cognitive RIS-aided NOMA network utilized to evaluate OP and asymptotic OP. **Lemma 1**. : _The corresponding PDF and CDF of the \(m\)-th ordered \(\mathcal{G}_{m}^{2}\) is given by_  
\[f_{\mathcal{G}_{m}^{2}}(x) =\sum_{a=0}^{M-m}\Delta_{M}(-1)^{a}f_{\mathcal{G}_{m}^{2}}(x) \Big{[}F_{\mathcal{G}_{m}^{2}}(x)\Big{]}^{m+a-1} \tag{5}\] \[F_{\mathcal{G}_{m}^{2}}(x) =\sum_{a=0}^{M-m}\frac{\Delta_{M}(-1)^{a}}{m+a}\Big{[}F_{ \mathcal{G}_{m}^{2}}(x)\Big{]}^{m+a} \tag{6}\]  
_where \(\Delta_{M}=\frac{M!}{(m-1)!(M-m-a)!a! }\)._  
Figure 1: A cognitive RIS-aided NOMA network. **Proof of Lemma 1. ** Following Rayleigh fading, the PDF and CDF, respectively, of \(|h_{j}|\), where \(j\in\{\mathcal{SR}_{n},\mathcal{R}_{n}\mathcal{U}_{m},\mathcal{SU}_{m},\mathcal{ SD}_{k}\}\), are  
\[f_{|h_{j}|}(x)=\frac{2x}{\Omega_{j}}e^{-\frac{x^{2}}{\Omega_{j}}},F_{|h_{j}|}(x)=1-e^{-\frac{x^{2}}{\Omega_{j}}}. \tag{7}\]  
where \(\Omega_{j}\) denotes the average power of channel. Furthermore, the \(k\)-th moment of \(|h_{j}|\) is obtained with  
\[\mu_{h_{j}}(k)=\int\limits_{0}^{\infty}x^{k}f_{|h_{j}|}(x)dx=\left( \sqrt{\Omega_{j}}\right)^{k}\Gamma\bigg{(}\frac{k}{2}+1\bigg{)} \tag{8}\]  
Then, the PDF expression for \(G_{m,n}\) can can be calculated as follows:  
\[f_{G_{m,n}}(x) =\int\limits_{0}^{\infty}\frac{1}{\gamma}f_{|h_{R}|_{dR}|_{m}} \bigg{(}\frac{x}{\gamma}\bigg{)}f_{|h_{SR}|_{n}}(\gamma)d\gamma \tag{9}\] \[=\frac{4x}{\Omega_{\mathcal{R}_{n}\mathcal{U}_{m}}\Omega_{ \mathcal{SR}_{n}}}\int\limits_{0}^{\infty}\frac{1}{\gamma}e^{-\frac{x^{2}}{ \Omega_{R}|_{dR}|_{m}}\gamma^{2}}e^{-\frac{\gamma^{2}}{\Omega_{SR_{n}}}}d\gamma. \tag{10}\]  
Using [Equation 3.471.9] in [46], the exact PDF of \(G_{m,n}\) is  
\[f_{G_{m,n}}(x)=\frac{4x}{\Omega_{\mathcal{R}_{n}\mathcal{U}_{m}} \Omega_{\mathcal{SR}_{n}}}K_{0}\bigg{(}\sqrt{\frac{4x^{2}}{\Omega_{\mathcal{R} _{n}\mathcal{U}_{m}}\Omega_{\mathcal{SR}_{n}}}}\bigg{)} \tag{11}\]  
where \(K_{0}(x)\) denotes the zero-th order modified Bessel function of the second kind [46]. With the help of the PDF of \(G_{m,n}\) and [Equation 6.561.16] in [46], the \(k\)-th moment of \(G_{m,n}\) is derived as  
\[F_{\mathcal{G}_{m}^{2}}(x) =\sum_{a=0}^{M-m}\frac{\Delta_{M}(-1)^{a}}{m+a}\Big{[}F_{ \mathcal{G}_{m}^{2}}(x)\Big{]}^{m+a} \tag{6}\]0  
With the help of the multinomial expansion [47], the \(k\)-th moment of \(G_{m}\) can be expressed as  
\[\mu_{G_{m}}(k) =\sum\limits_{k_{1}=0}^{k}\sum\limits_{k_{2}=0}^{k_{1}}\cdots \sum\limits_{k_{N-1}}^{k_{N-2}}\left( \begin{array}{c}k\\ k_{1}\end{array} \right)\bigg{(} \begin{array}{c}k_{1}\\ k_{2}\end{array} \bigg{)}\cdots\left( \begin{array}{c}k_{N-2}\\ k_{N-1}\end{array} \right)\] \[\times\mu_{G_{m,1}}(k-k_{1})\mu_{G_{m,2}}(k_{1}-k_{2})\cdots\mu_{ G_{m,N}}(k_{N-1}) \tag{13}\]  
Based on the independence of \(G_{m}\) and \(h_{\mathcal{SU}_{m}}\), with the help of [Equation (1.111)] in [46], the \(k\)-th moment of unordered \(\bar{Q}_{m}^{2}\) can be expressed as  
\[\mu_{G_{m}^{2}}(k)=\sum\limits_{q=0}^{2k}\left( \begin{array}{c}2k\\ q\end{array}