# Performance Analysis of a Cognitive RIS-NOMA in Wireless Sensor Network
## 3 Performance Analysis
### Outage Probability Analysis  
Before the \(m\)-th SU decodes its own signal, the SIC is executed at that SU by identifying and canceling the information of the p-th SU (\(m\geqslant p\)). An outage occurs if the \(m\)-th SU is unable to identify the information of the \(p\)-th SU, as indicated by  
\[E_{m,p}=\left\{\frac{\mathcal{G}_{m}^{2}P_{S}\beta_{p}}{\mathcal{G}_{m}^{2}P_ {S}\sum_{i=p+1}^{M}\beta_{i}+\zeta\sigma^{2}+\sigma^{2}}>\gamma_{p}\right\} \tag{18}\]  
where \(\gamma_{p}=2^{R_{p}}-1\), \(R_{p}\) denotes the target rate of \(m\)-th SU to detect \(x_{p}\). Then, the outage probability of \(m\)-th SU is given by  
\[\mathcal{P}_{m}=1-\Pr[E_{m,1}\cap E_{m,2}\cap\ldots\cap E_{m,m}] \tag{19}\]  
**Theorem 1**. : _The approximate expression OP of \(m\)-th SU is given by_  
\[\mathcal{P}_{m} \approx\sum_{l=0}^{K}\sum_{a=0}^{M-m}\left( \begin{array}{c}K\\ l\end{array} \right)\frac{\Delta_{M}(-1)^{l+a}e^{-\frac{l\eta_{th}}{l_{SP} \beta_{s}}}}{m+a}\bigg{[}\frac{1}{\Gamma(k_{m})}\gamma\bigg{(}k_{m},\frac{ \theta_{m}^{*}}{\eta_{S}w_{m}}\bigg{)}\bigg{]}^{m+a}\] \[+\frac{K\pi^{2}}{\Omega_{SP}4l}\sum_{l=0}^{K-1}\sum_{a=0}^{M-m} \left( \begin{array}{c}K-1\\ l\end{array} \right)\frac{\Delta_{M}(-1)^{l+a}}{(m+a)}\] \[\times\sum_{c=1}^{l}\sqrt{1-\varphi_{c}^{2}}\sec^{2}\bigg{(} \frac{(\varphi_{c}+1)\pi}{4}\bigg{)}\bigg{[}\frac{1}{\Gamma(k_{m})}\gamma \bigg{(}k_{m},\frac{\theta_{m}^{*}\partial_{c}}{w_{m}\eta_{th}}\bigg{)} \bigg{]}^{m+a}e^{-\frac{l+1}{l_{SD}}\omega_{c}} \tag{20}\]  
**Proof of Theorem 1. ** With the help of (1) and (18), the OP in (19) can be rewritten as  
\[\mathcal{P}_{m} =\Pr\biggl{[}\mathcal{G}_{m}^{2}<\frac{\theta_{m}^{*}}{\eta_{ \mathcal{S}}},Z_{\mathcal{SD}}<\frac{\eta_{th}}{\eta_{\mathcal{S}}}\biggr{]}+ \Pr\biggl{[}\mathcal{G}_{m}^{2}<\frac{\theta_{m}^{*}Z_{\mathcal{SD}}}{\eta_{th }},Z_{\mathcal{SD}}>\frac{\eta_{th}}{\eta_{\mathcal{S}}}\biggr{]}\] \[=\underbrace{\int\limits_{0}^{\frac{\eta_{th}}{\eta_{\mathcal{S} }}}\int\limits_{0}^{\frac{\eta_{th}}{\eta_{\mathcal{S}}}}(x)dxdy}_{I_{1}}+ \underbrace{\int\limits_{\frac{\eta_{th}}{\eta_{\mathcal{S}}}}^{\infty}f_{Z_{ \mathcal{SD}}}(y)}_{\frac{\eta_{th}}{\eta_{\mathcal{S}}}}(y)\int\limits_{0}^{ \frac{\eta_{th}}{\eta_{\mathcal{S}}}}(x)dxdy}_{I_{2}} \tag{21}\]  
where \(\eta_{\mathcal{S}}=\frac{\bar{p}_{\mathcal{S}}}{\sigma^{2}}\), \(\eta_{th}=\frac{P_{th}}{\sigma^{2}}\), \(\theta_{m}^{*}=\max(\vartheta_{1},\ldots,\vartheta_{m})\), \(\theta_{m}=\frac{\gamma_{m}(\zeta+1)}{\left(\beta_{m}-\gamma_{m}\sum\limits_{ i=m+1}^{M}\hat{p}_{i}\right)}\), \(\theta_{M}=\frac{\gamma_{m}(\zeta+1)}{\beta_{M}}\), and \(Z_{\mathcal{SD}}=\max\limits_{k=1,\ldots,K}\bigl{|}h_{\mathcal{SD}_{k}}\bigr{|} ^{2}\). Based on [Equations (19) and (20)] in [48], the PDF and CDF of \(Z_{\mathcal{SD}}\) are expressed, respectively, as  
\[f_{Z_{\mathcal{SD}}}(x) =\sum\limits_{l=0}^{K-1}\left( \begin{array}{c}K-1\\ l\end{array} \right)\frac{(-1)^{l}K}{\Omega_{\mathcal{SD}}}e^{-\frac{l+1}{lS \mathcal{D}}x} \tag{22}\] \[F_{Z_{\mathcal{SD}}}(x) =\sum\limits_{l=0}^{K}\left( \begin{array}{c}K\\ l\end{array} \right)(-1)^{l}e^{-\frac{l}{lS\mathcal{D}}x} \tag{23}\]  
Furthermore, the term \(I_{1}\) can be obtained as  
\[I_{1} =F_{Z_{\mathcal{SD}}}\biggl{(}\frac{\eta_{th}}{\eta_{\mathcal{S} }}\biggr{)}F_{\mathcal{G}_{m}^{2}}\biggl{(}\frac{\theta_{m}^{*}y}{\eta_{ \mathcal{S}}}\biggr{)}\] \[=\sum\limits_{l=0}^{K}\sum\limits_{a=0}^{M-m}\left( \begin{array}[] {c}K\\ l\end{array} \right)\frac{\Delta_{M}(-1)^{l+a}e^{-\frac{l\eta_{th}}{\eta_{ \mathcal{SD}}\eta_{\mathcal{S}}}}}{m+a}\left[\frac{1}{\Gamma(k_{m})}\gamma \biggl{(}k_{m},\frac{\theta_{m}^{*}}{\eta_{\mathcal{S}}w_{m}}\biggr{)} \right]^{m+a} \tag{24}\]  
Next, the term \(I_{2}\) in (21) is calculated as  
\[I_{2} =\int\limits_{\frac{\eta_{th}}{\eta_{\mathcal{S}}}}^{\infty}f_{Z _{\mathcal{SD}}}(y)F_{\mathcal{G}_{m}^{2}}\biggl{(}\frac{\theta_{m}^{*}y}{ \eta_{th}}\biggr{)}dy\] \[=\sum\limits_{l=0}^{K-1}\sum\limits_{a=0}^{M-m}\left( \begin{array} []{c}K-1\\ l\end{array}