# Hybrid Cooperative Spectrum Sharing Protocols Based on OMA and NOMA in Cognitive Radio Networks
## Appendix B  
**Proof of Theorem 2.** In the underlay OMA mode, the end-to end achieved rate of the channel PT \(\rightarrow\) PR is expressed as \(R_{\text{PT}\rightarrow\text{PR},\gamma_{p}}^{\text{underlay}}=log_{2} \Big{(}1+\frac{|h_{PP}|^{2}P_{P}}{|h_{SP}|^{2}P_{S}+\sigma^{2}}\Big{)}\). The outage probability of this channel with SU's interference is expressed as:  
\[\begin{split}& P_{\text{PT}\rightarrow\text{PR}}^{\text{out,I}}=Pr \Big{(}R_{\text{PT}\rightarrow\text{PR},\gamma_{P}}^{\text{underlay}}<R_{P} \Big{)}=Pr\Big{(}log_{2}\bigg{(}1+\frac{|h_{PP}|^{2}P_{P}}{|h_{SP}|^{2}P_{S}+ \sigma^{2}}\bigg{)}<R_{P}\Big{)}\\ &=Pr\Big{(}|h_{PP}|^{2}<\frac{\varphi_{P}}{\rho_{P}}\Big{(}|h_{SP }|^{2}\rho_{S}+1\Big{)}\Big{)}=Pr\Big{(}|h_{PP}|^{2}<\frac{\varphi_{P}}{\rho_ {P}}\Big{(}|h_{SP}|^{2}min\Big{\{}\frac{\rho_{th}}{|h_{SP}|^{2}},\rho_{1} \Big{\}}+1\Big{)}\Big{)}\\ &=Pr\bigg{(}\frac{\rho_{th}}{|h_{SP}|^{2}}<\rho_{1},|h_{PP}|^{2}< \frac{\varphi_{P}}{\rho_{P}}\Big{(}|h_{SP}|^{2}\frac{\rho_{th}}{|h_{SP}|^{2}} +1\Big{)}\bigg{)}+Pr\Big{(}\frac{\rho_{th}}{|h_{SP}|^{2}}>\rho_{1},|h_{PP}|^{2 }<\frac{\varphi_{P}}{\rho_{P}}\Big{(}|h_{SP}|^{2}\rho_{1}+1\Big{)}\bigg{)}\\ &=Pr\Big{(}|h_{SP}|^{2}>\frac{\rho_{th}}{\rho_{1}},|h_{PP}|^{2}< \frac{\varphi_{P}}{\rho_{P}}(\rho_{th}+1)\Big{)}+Pr\Big{(}|h_{SP}|^{2}<\frac {\rho_{th}}{\rho_{1}},|h_{PP}|^{2}<\frac{\varphi_{P}}{\rho_{P}}\Big{(}|h_{SP}| ^{2}\rho_{1}+1\Big{)}\Big{)}\\ &=Q_{1}+Q_{2},\end{split} \tag{2}\]  
\[\begin{split}&\text{where }\varphi_{S}=2^{R_{S}}-1,\rho_{1}=\frac{p_{ \text{max}}}{\sigma^{2}},\rho_{th}=\frac{I_{th}}{\sigma^{2}},\text{ and }Q_{1}\text{ is expressed as}\\ &\text{}Q_{1}=Pr\Big{(}|h_{SP}|^{2}>\frac{\rho_{th}}{\rho_{1}},|h_{ PP}|^{2}<\frac{\varphi_{P}(\rho_{th}+1)}{\rho_{P}}\Big{)}=\int_{\frac{\rho_{th}}{ \rho_{1}}}^{\infty}\int_{0}^{\frac{\varphi_{P}(\rho_{th}+1)}{\rho_{P}}}\frac{1 }{\lambda_{PP}}exp\Big{(}-\frac{x}{\lambda_{PP}}\Big{)}e^{-\frac{y}{\lambda_{ SP}}}dxdy\\ &=\Big{(}1-exp\Big{(}-\frac{\varphi_{P}(\rho_{th}+1)}{\lambda_{ SPP}\rho_{1}}\Big{)}\Big{)}exp\Big{(}-\frac{\rho_{th}}{\lambda_{SP}\rho_{1}}\Big{)},\end{split} \tag{3}\]  
and \(Q_{2}\) is expressed as\[\begin{array}{l}Q_{2}=Pr\Big{(}|h_{SP}|^{2}<\frac{\rho_{th}}{\rho_{1}},|h_{SP}|^{ 2}<\frac{\varphi_{p}}{\rho_{p}}\Big{(}|h_{SP}|^{2}\rho_{1}+1\Big{)}\Big{)}=\int_ {0}^{\frac{\rho_{th}}{\rho_{1}}}\int_{0}^{\frac{\rho_{p}(\varphi_{p}+1)}{\rho_ {p}}}\frac{1}{\lambda_{pp}}e^{-\frac{x}{\lambda_{pp}}}\frac{1}{\lambda_{SP}}e^{ -\frac{y}{\lambda_{SP}}}dxdy\\ =\int_{0}^{\frac{\rho_{th}}{\rho_{1}}}\frac{1}{\lambda_{SP}}e^{-\frac{y}{ \lambda_{SP}}}\Big{(}1-e^{-\frac{\varphi_{p}(\varphi_{p}+1)}{\rho_{p}\lambda_{ SP}}}\Big{)}dy=\int_{0}^{\frac{\rho_{th}}{\rho_{1}}}\frac{1}{\lambda_{SP}}e^{- \frac{y}{\lambda_{SP}}}dy-\int_{0}^{\frac{\rho_{th}}{\rho_{1}}}\frac{1}{\lambda _{SP}}e^{-\frac{(\frac{\varphi_{p}(\varphi_{p}+1)}{\rho_{p}\lambda_{SP}}+\frac{ y}{\lambda_{SP}})}}dy\\ =1-exp(-\frac{\rho_{th}}{\rho_{1\lambda_{SP}}})-exp(-\frac{\varphi_{p}}{\rho_ {2\lambda_{SP}}})\int_{0}^{\frac{\rho_{th}}{\rho_{1}}}\frac{1}{\lambda_{SP}}e ^{-\frac{(\frac{\varphi_{p}\rho_{1}\lambda_{SP}}+\varphi_{p}\lambda_{SP})}{ \rho_{p}\lambda_{SP}}y}dy\\ =1-exp\Big{(}-\frac{\rho_{th}}{\rho_{1\lambda_{SP}}}\Big{)}-\frac{\rho_{p} \lambda_{SP}}{\varphi_{p}\lambda_{SP}+\varphi_{p}\lambda_{SP}}exp(-\frac{ \varphi_{p}}{\rho_{AP}})\Big{(}1-exp\Big{(}-\frac{\varphi_{p}\rho_{th}}{\rho_ {AP}}-\frac{\rho_{th}}{\rho_{1\lambda_{SP}}}\Big{)}\Big{)}.\end{array} \tag{10}\]