# Hybrid Cooperative Spectrum Sharing Protocols Based on OMA and NOMA in Cognitive Radio Networks
## 2 System Model and Protocol Descriptions
### Throughput of SUs
The SU's throughput for hybrid cooperative cognitive radio network base on NOMA and OMA can be written as:  
\[ \begin{array}{l}\Omega_{\text{SU}}^{\frac{OMA}{\text{SU}}}=\left(1-P_{\text{ ST}\to\text{SR}}^{out,x_{\text{S}},t}\right)\pi_{S_{2}}+\Big{(}1-P_{\text{ ST}\to\text{SR}}^{out,x_{\text{S}}}\Big{)}\pi_{S_{3}}\\ =\frac{\Big{(}1-P_{\text{ST}\to\text{SR}}^{out,x_{\text{S}}}\Big{)}P_{1}}{1+2P_ {1}+P_{4}+P_{3}-P_{1}P_{2}}+\frac{\big{(}1-P_{\text{ST}\to\text{SR}}^{out,x_{ \text{S}}}\big{)}P_{4}}{1+2P_{1}+P_{4}+P_{3}-P_{1}P_{2}}\\ =\frac{\Big{(}1-P_{\text{ST}\to\text{SR}}^{out,x_{\text{S}}}\Big{)}P_{1}+\big{(} 1-P_{\text{ST}\to\text{SR}}^{out,x_{\text{S}}}\big{)}P_{4}}{1+2P_{1}+P_{4}+P_ {3}-P_{1}P_{2}}=\frac{E}{A}\,\end{array}  \tag{31}\]  
where the closed-form of \(A\) is the same as Equation (26), and subsisting Equations (24), (29) and (30), into Equation (31), we can obtain the closed-form of \(E\) is related to the relationship between \(\alpha_{P}/\alpha_{S}\) and \(\varphi_{P}\), i.e.,  
(1) For \(\alpha_{P}/\alpha_{S}\geq\varphi_{P}\), we have:  
\[ \begin{array}{l}E=\Big{(}\frac{\rho_{1}\lambda_{SS}}{\rho_{1}\lambda_{SS}+ \sigma_{SP}\rho_{PR}}\Big{(}1-exp\Big{(}-\frac{\rho_{h}}{\rho_{1}\lambda_{SS}} \Big{)}\Big{)}+\frac{\rho_{h}\lambda_{SS}}{\varphi_{S}\lambda_{SP}\rho_{AP}} exp\Big{(}-\frac{\rho_{h}}{\rho_{1}\lambda_{SP}}+DC\Big{)}\cdot\text{Ei}(-CD) \Big{)}exp\Big{(}-\frac{\varphi_{P}}{\rho_{P}\lambda_{PP}}-\frac{\varphi_{S}}{ \rho_{1}\lambda_{SS}}\Big{)}\\ +\Big{(}1-exp\Big{(}-\frac{\varphi_{P}}{\rho_{P}\lambda_{PP}}\Big{)}\Big{)} exp\Big{(}-\frac{\varphi_{P}}{\rho_{P}\lambda_{PS}}-\frac{\varphi}{\lambda_{SS}} \Big{)},\end{array}  \tag{32}\]  
\[ \begin{array}{l}\text{where }\phi\!=\max\Big{\{}\frac{\varphi_{P}}{(\alpha_{P}- \alpha_{S}\varphi_{P})\rho_{S}},\frac{\varphi_{S}}{\alpha_{S}\rho_{S}}\Big{\}}.\\ \text{(2)}\quad\text{For }\alpha_{P}/\alpha_{S}\geq\varphi_{P}\text{, we have:}\end{array}  \tag{33}\]  
Throughput Analysis of Primary and Secondary User in Hybrid Cooperative Cognitive Radio Network Based on Pure OMA  
In the section, the throughputs of the PU and SU for a hybrid cooperative cognitive radio network based on pure OMA are derived. In this case, the transmission of non-CR transmission mode and underlay OMA transmission mode are the same as that in the situation of OMA and NOMA. Therefore, only the part of the overlay OMA cooperative mode is analyzed here. In overlay OMA cooperative mode, data transmission can be divided into two cases:1. ST can decode the PU's packet and act as a relay to help forward the PU's packet to PR. The achieved date rate for ST to decode the PU's packet and the channel ST \(\rightarrow\) PR are given as: \[R_{\text{PT}\rightarrow\text{ST},x_{p}}^{\text{overlay,OMA}} =log_{2}\Bigg{(}1+\frac{|h_{PS}|^{2}P_{P}}{\sigma^{2}}\Bigg{)},\] (34) \[R_{\text{ST}\rightarrow\text{PR},x_{p}}^{\text{overlay,OMA}} =log_{2}\Bigg{(}1+\frac{|h_{SP}|^{2}P_{S}}{\sigma^{2}}\Bigg{)},\] (35) 2. ST fails to decode the PU's packet, PT will retransmit that the PU's packet failed previously while SU keeps silent.