# Hybrid Cooperative Spectrum Sharing Protocols Based on OMA and NOMA in Cognitive Radio Networks
## 2 System Model and Protocol Descriptions
### Throughput of SUs  
In this section, the throughput of the SU is derived for hybrid cooperative cognitive radio network base on NOMA and OMA. In this network, the SU's packet is transmitted over the channel ST \(\rightarrow\) SR under the interference of PT's transmission in underlay OMA transmission mode, and ST transmits a superimposed signal to SR in overlay NOMA cooperative mode. In the overlay NOMA mode, for the successful transmission of SUs packet in channel ST \(\rightarrow\) SR, SR can decode its own packet only if it successfully decodes PUs packets. The outage probability of the channel ST \(\rightarrow\) SR is given by:  
\[ \begin{array}{l}P_{{\rm ST}\rightarrow{\rm SR}}^{out,x_{S}}=1-Pr\Big{(}R_{{ \rm ST}\rightarrow{\rm SR},x_{P}}^{\rm overlay,NOMAMA}\geq R_{P},R_{{\rm ST} \rightarrow{\rm SR},x_{S}}^{\rm overlay,NOMA}\geq R_{S}\Big{)}\\ =1-Pr\Big{(}log_{2}\Big{(}1+\frac{|h_{{\rm S}}|^{2}a_{p}P_{5}}{|h_{{\rm S}}|^{ \infty}s_{{\rm S}}P_{5}+\sigma^{2}}\Big{)}\geq R_{P},log_{2}\Big{(}1+\frac{|h_{ {\rm S}}|^{2}a_{p}P_{5}}{\sigma^{2}}\Big{)}\geq R_{S}\Big{)}\\ =1-Pr\Big{(}log_{2}\Big{(}1+\frac{|h_{{\rm S}}|^{2}a_{p}P_{5}}{|h_{{\rm S}}|^{ \infty}s_{{\rm S}}P_{5}+\sigma^{2}}\Big{)}\geq R_{P}\Big{)}\cdot Pr\Big{(}log_{2 }\Big{(}1+\frac{|h_{{\rm S}}|^{2}a_{p}P_{5}}{\sigma^{2}}\Big{)}\geq R_{S}\Big{)} \\ =1-Pr\Big{(}|h_{{\rm S}}|^{2}\geq\frac{\varphi_{p}}{(a_{p}-a_{p}\varphi_{{\rm P}} \lambda_{{\rm P}})\rho_{{\rm S}}}\Big{)}\cdot Pr\Big{(}|h_{{\rm S}}|^{2}\geq \frac{\varphi_{S}}{\alpha_{{\rm S}}\rho_{{\rm S}}}\Big{)}\\ =\Bigg{\{}1-\int_{\phi}^{\infty}\frac{1}{\lambda_{{\rm S}}\xi}e^{-\frac{\varphi_{S }}{\lambda_{{\rm S}}\xi}}dx,\begin{array}{l}\frac{\alpha_{P}}{\alpha_{S}} \geq\varphi_{P}\\ \frac{\alpha_{P}}{\alpha_{S}}<\varphi_{P}\end{array}\Bigg{\{}1-exp\Big{(}- \frac{\phi}{\lambda_{{\rm S}}\xi}\Big{)},\begin{array}{l}\frac{\alpha_{P}}{ \alpha_{S}}\geq\varphi_{P}\\ \frac{\alpha_{P}}{\alpha_{S}}<\varphi_{P}\end{array}\Bigg{\}}\\ \mbox{where }\phi=\max\Big{\{}\frac{\varphi_{p}}{(a_{p}-a_{S}\varphi_{p})\rho_{{\rm S}}}, \frac{\sigma_{S}}{\alpha_{S}\rho_{{\rm S}}}\Big{\}}.\end{array} \]  
**Theorem 3**. : _In the underlay OMA mode, the SU's packet is transmitted over the channel \(ST\to SR\) under the interference of PT's transmission. The outage probability of \(ST\to SR\) channel is given by:_  
\[P_{\text{ST}\to SR}^{out,x_{\text{S}},t}=Pr\Big{(}R_{\text{ST}\to\text{SR},x_{ \text{S}}}^{\text{underlay}}<R_{S}\Big{)}=Q_{3}+Q_{4} \tag{30}\]  
_where \(Q_{3}\)= \(exp\Big{(}-\frac{\rho_{h}}{\rho_{1}\lambda_{SS}}\Big{)}\) - \(\frac{\rho_{h}\lambda_{SS}}{\rho_{5}\rho_{5}\lambda_{SP}\rho_{R}}\cdot exp \Big{(}-\frac{\varphi_{S}}{\rho_{1}\lambda_{SS}}-\frac{\rho_{h}}{\rho_{1} \lambda_{SP}}\Big{)}\cdot exp(DC)\cdot Ei(-CD)\), \(C=\frac{\varphi_{S}\rho_{5}\lambda_{SP}+\lambda_{SS}\rho_{1}\lambda_{PR}}{ \lambda_{SS}\lambda_{SP}\rho_{1}\lambda_{PR}},D=\frac{\varphi_{S}\lambda_{SP} +\rho_{h}\lambda_{SS}}{\varphi_{SP}\rho_{Step}}\), and \(Q_{4}=\Big{(}1-exp\Big{(}-\frac{\rho_{h}}{\rho_{1}\lambda_{SP}}\Big{)}\Big{)}\) \(\Big{(}1-\frac{\rho_{1}\lambda_{SS}}{\rho_{1}\lambda_{SS}+\sigma_{SP}\rho_{ PR}}\cdot exp\Big{(}-\frac{\varphi_{S}}{\rho_{1}\lambda_{SS}}\Big{)}\Big{)}\)._  
Proof. : See Appendix C.  
In the protocol for OMA and NOMA technology, the SU can transmit its own packet at the state \(S_{2}\) and \(S_{3}\), according to the Markov model given in Figure 3. If PR sends ACK feedback in the non-CR transmission mode, the proposed protocol moves on the underlay OMA mode, which corresponds to the state \(S_{2}\) and the SU transmits under the interference of the PU's transmission. The packet success probability is equal to \(1-P_{\text{ST}\to SR}^{out,x_{\text{S}},t}\). If a NACK is received from PR in the non-CR transmission mode and ST can successfully decode the PU's packet, the protocol operates at the state \(S_{3}\). The success probability of the SU's packet is equal to \(1-P_{\text{ST}\to SR}^{out,x_{\text{S}}}\).