# Hybrid Cooperative Spectrum Sharing Protocols Based on OMA and NOMA in Cognitive Radio Networks
## 2 System Model and Protocol Descriptions
### Throughput of PUs  
In this subsection, the throughput of the PU for hybrid cooperative cognitive radio network based on OMA and NOMA is derived. The throughput is defined as the packet \begin{table}
\begin{tabular}{c c c} \hline \hline
**State** & **PT** & **ST** \\ \hline S\({}_{1}\) & PU’s packet & Silent \\ S\({}_{2}\) & PU’s packet & SU’s packet \\ S\({}_{3}\) & Silent & PU’s packet retransmission \\ S\({}_{4}\) & PU’s packet retransmission & Silent \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Hybrid cooperative cognitive radio networks based on pure OMA of transmission state. transmission success probability multiplied by the steady-state probability. The probability that the target rate exceeds the instantaneous achievable rate of the channel at certain time is defined as the outage probability. The outage probability between any two non-interfering nodes in the channel is given by:  
\[p_{j}^{out}=1-exp\bigg{(}-\frac{\varphi_{P}}{\lambda_{i}\varphi_{P}}\bigg{)}, \tag{21}\]  
where \(\varphi_{P}=2^{R_{P}}-1\), \(\rho_{P}=\frac{P_{P}}{\sigma^{2}}\), \(j=\mathrm{PT}\rightarrow\mathrm{PR},\mathrm{PT}\rightarrow\mathrm{ST}\), and \(i=PP,PS\). Proof. : See Appendix A. In the underlay OMA mode, PT transmits packet of PU under SU's interference. The outage probability of \(PT\to PR\) channel is expressed as:  
\[P_{PT\to PR}^{out,I}=\Q_{1}+\Q_{2}, \tag{22}\]  
where \(\Q_{2}=1-exp\bigg{(}-\frac{\rho_{lb}}{\rho_{1}\lambda_{SP}}\bigg{)}-\frac{ \rho_{P}\lambda_{SP}}{\varphi_{P}\rho_{1}\lambda_{SP}+\rho_{P}\lambda_{SP}}exp \bigg{(}-\frac{\varphi_{P}}{\rho_{P}\lambda_{SP}}\bigg{)}\bigg{(}1-exp\bigg{(} -\frac{\varphi_{P}\rho_{lb}}{\rho_{P}\lambda_{SP}}-\frac{\rho_{lb}}{\rho_{1} \lambda_{SP}}\bigg{)}\bigg{)}\), and \(\Q_{1}=\Big{(}1-exp\bigg{(}-\frac{\varphi_{P}\rho_{lb}}{\lambda_{PP}\rho_{P}}- \frac{\varphi_{P}}{\lambda_{PP}\rho_{P}}\bigg{)}\bigg{)}exp\bigg{(}-\frac{ \rho_{P}}{\rho_{P}\lambda_{SP}}\bigg{)}\), with \(\rho_{1}=\frac{P_{max}}{\sigma^{2}}\), \(\rho_{th}=\frac{I_{lb}}{\sigma^{2}}\), and \(\varphi_{P}=2^{R_{P}}-1\). Proof. : See Appendix B. In the overlay NOMA mode, ST transmits superimposed signal \(\sqrt{\alpha_{S}P_{S}}x_{S}+\sqrt{\alpha_{P}P_{S}}x_{P}\) with \(\alpha_{S}+\alpha_{P}=1\) to PR and SR simultaneously. The outage probability of \(\mathrm{ST}\rightarrow\mathrm{PR}\) is expressed as:  
\[P_{\mathrm{ST}\rightarrow\mathrm{PR}}^{out,\mathrm{NOMA}}=Pr \Big{(}R_{\mathrm{ST}\rightarrow\mathrm{PR},x_{P}}^{\mathrm{overlap}, \mathrm{NOMA}}<R_{P}\Big{)}=Pr\bigg{(}log_{2}\bigg{(}1+\frac{|h_{SP}|^{2} \alpha_{P}P_{S}}{|h_{SP}|^{2}\alpha_{S}P_{S}+\sigma^{2}}\bigg{)}<R_{P}\Big{)} \tag{23}\] \[=Pr\Big{(}|h_{SP}|^{2}<\frac{\varphi_{P}}{\alpha_{P}\rho_{S}- \varphi_{P}\alpha_{S}\rho_{S}}\Big{)}= \begin{cases}\int_{0}^{\frac{\varphi_{P}} {\alpha_{P}\rho_{S}-\varphi_{P}\alpha_{S}\rho_{S}}}\frac{1}{\lambda_{SP}}e^{- \frac{\pi}{\lambda_{SP}}}dx,&\frac{\alpha_{P}}{\alpha_{S}}\geq\varphi_{P}\\ 1,&\frac{\alpha_{P}}{\alpha_{S}}<\varphi_{P}\end{cases} \] \[= \begin{cases}1-exp(-\frac{\varphi_{P}}{\alpha_{P}\rho_{S}\lambda_ {SP}-\varphi_{P}\alpha_{S}\rho_{S}\lambda_{SP}}),&\frac{\alpha_{P}}{\alpha_{S} }\geq\varphi_{P}\\ 1,&\frac{\alpha_{P}}{\alpha_{S}}<\varphi_{P}\end{cases} \]  
The PU's throughput for hybrid cooperative radio wireless network base on NOMA and OMA is derived by the steady state probabilities and state transition probabilities of the Markov chain, given in Figure 3. The outage probabilities of the corresponding channels are given by Equations (21)-(23).