# Achievable Rate of NOMA-Based Cooperative Spectrum-Sharing CRN over Nakagami-\(m\) Channels
## 4 Achievable Rate Analysis
### Preliminaries  
Let \(h\) denote the Nakagami-\(m\)-distributed fading channel coefficient with the probability density function (PDF) as ([22], Equation (5.35))  
\[f_{h}(h)=\frac{2m^{m}h^{2m-1}}{\Omega^{m}\Gamma(m)}exp\biggl{(}-\frac{mh^{2} }{\Omega}\biggr{)},\ \ \ \ h\geq 0, \tag{11}\]  
where \(m\geq 0.5\) is the fading severity parameter, \(\Gamma(\cdot)\) is the Gamma function ([23], Equation (8.310.1)), and \(\Omega\)=\(\mathbb{E}\{|h|^{2}\}\). Nakagami-\(m\) distribution is generally used to model fading scenarios with one strong, i.e., line-of-sight (LoS), and many weaker, i.e., non line-of-sight signal, components. This distribution approximates the fading channel's coefficient of a vector process where the central limit theorem may not be necessarily valid. Moreover, when the fading channel coefficient of a certain link is Nakagami-\(m\)-distributed, then the instantaneous SNR for the corresponding link is gamma-distributed. The PDF of the instantaneous SNR can then be written as ([22], Equation (5.37)).  
\[f_{\gamma}(\gamma)=\frac{m^{m}\gamma^{m-1}}{\overline{\gamma}^{m}\Gamma(m)}exp\left( -\frac{m\gamma}{\overline{\gamma}}\right),\ \ \ \ \gamma\geq 0 \tag{12}\]  
where \(\gamma\)=\(\frac{P}{\sigma^{2}}|h|^{2}\) is the instantaneous SNR, and \(\overline{\gamma}\)=\(\frac{P}{\sigma^{2}}\Omega\) is the average SNR. It may be noted that \(P\) denotes the transmit power whereas \(\sigma^{2}\) represents the variance of the zero mean Gaussian thermal noise added at the receiver. This PDF expression can alternatively be written in terms of [PERSON]'s H-function as ([24], Equation (2)).  
\[f_{\gamma}(\gamma)=\frac{\left(\frac{m}{\gamma}\right)}{\Gamma(m)}H_{0,1}^{1,0 }\left[\frac{m\gamma}{\overline{\gamma}}\right|\ \ \frac{m\overline{\gamma}}{(m-1,1)}\ \ \right], \tag{13}\]  
where \(H_{p,q}^{m,n}[\cdot]\) is the [PERSON]'s H-function represented by Mellin-Barnes-type integrals and is defined as ([25], Equation (1.1.1)). [PERSON]'s H-function is a generalized function in the sense that most special functions of one variable can be represented in terms of this function. It has applications in various problems arising in engineering, physical sciences, and statistics [25].  
When the transmitter does not have knowledge of the channel state information (CSI), then it can not adapt its power in accordance to CSI. It is then feasible for the transmitter to operate with constant power. Therefore, the ergodic capacity of the fading channel is obtained by averaging over the SNR distribution and is defined as ([22], Equation (14.74)).  
\[\bar{R}:=\frac{1}{2}\int_{0}^{\infty}\log_{2}(1+\gamma)f_{\gamma}(\gamma)\,d \gamma, \tag{14}\]  
where factor \(\frac{1}{2}\) is a result of the two transmission phases. For mathematical convenience, function \(log_{2}(1+\gamma)\) in (14) can be expressed in terms of [PERSON]'s H-function ([26], Equation (2.4.3)). This representation of \(log_{2}(1+\gamma)\) will facilitate us in the upcoming calculations.  
\[\log_{2}(1+\gamma)=\frac{1}{\ln(2)}H_{2,2}^{1,2}\left[\gamma\right]\ \ \begin{pmatrix}(1,1),(1,1)\\ (1,1),(0,1)\end{pmatrix}. \tag{15}\]