## 3 Performance Analysis
### Capacity Analysis with Adaptive Transmission
#### 3.2.1 Optimal Power and Rate Adaptation (Opra)  
In OPRA, the node \(S\) adapts both the transmission rate as well as the transmission power in accordance with the channel conditions. The average channel capacity using this scheme is given by [2] (Equation (17)):  
\[\overline{C}_{OPRA}=\frac{B}{2}\int_{\gamma_{o}}^{\infty}\log_{2}\bigg{(}\frac {\gamma}{\gamma_{o}}\bigg{)}f_{\gamma_{I}}(\gamma)d\gamma, \tag{7}\]  
where \(\gamma_{o}\) and the parameter \(B\) denote an optimal cutoff SNR level and a channel bandwidth in Hertz, respectively. The transmission between the node \(S\) and node \(D\) is halted when the SNR level drops below the cutoff SNR level \(\gamma_{o}\). Therefore, it must satisfy [2]:  
\[\int_{\gamma_{o}}^{\infty}\bigg{(}\frac{\gamma-\gamma_{o}}{\gamma\gamma_{o}} \bigg{)}f_{\gamma_{I}}(\gamma)d\gamma=1. \tag{8}\]To obtain the average capacity for OPRA, \(\overline{C}_{OPRA}\), node \(S\) and node \(D\) must have a perfect knowledge of CSI [14]. In line with the channel conditions, node \(S\) adjusts its transmit rate and transmit power, such that higher rates and powers are assigned to good channel conditions and _vice versa_. In this technique, the instantaneous adaptive power and instantaneous adaptive rate, are, respectively, given by [2]:  
\[P_{OPRA}/P=\left\{\begin{array}{ll}0,&\gamma<\gamma_{o}\\ \frac{1}{\gamma_{o}}-\frac{1}{\gamma},&\gamma\geq\gamma_{o}\end{array}\right. \tag{9}\]  
and  
\[R_{OPRA}=\frac{B}{2}\log_{2}\left(\frac{\gamma}{\gamma_{o}}\right), \tag{10}\]  
where \(P\) designates the average power constraint. By employing the logarithmic properties, Equation (7) can be rewritten as:  
\[\overline{C}_{OPRA}=\frac{B}{2\ln(2)}\int_{\gamma_{o}}^{\infty}\bigg{[}\ln( \gamma)-\ln(\gamma_{o})\bigg{]}f_{\gamma_{l}}(\gamma)d\gamma. \tag{11}\]  
Inserting Equation (6) into Equation (11), we have:  
\[\begin{split}\overline{C}_{OPRA}=\sum_{i=0}^{\infty}\sum_{j=0}^ {\infty}&\sum_{n=0}^{N_{1}\mu_{1}+i-1}\sum_{l=0}^{N_{2}\mu_{2}+j-1} \frac{B\xi_{1}\xi_{2}\Delta\Pi\Psi}{2\ln(2)}\bigg{[}\bigg{\{}\int_{\gamma_{o} }^{\infty}\frac{\ln(\gamma)\gamma^{n+l}(\psi_{1}+\psi_{2})}{\varepsilon^{( \psi_{1}+\psi_{2})\gamma}}d\gamma-\int_{\gamma_{o}}^{\infty}\frac{\ln(\gamma) \gamma^{n+l-1}(n+l)}{\varepsilon^{(\psi_{1}+\psi_{2})\gamma}}d\gamma\bigg{\}} \\ &-\gamma_{o}\bigg{\{}\int_{\gamma_{o}}^{\infty}\frac{\gamma^{n+l} (\psi_{1}+\psi_{2})}{\varepsilon^{(\psi_{1}+\psi_{2})\gamma}}d\gamma-\int_{ \gamma_{o}}^{\infty}\frac{\gamma^{n+l-1}(n+l)}{\varepsilon^{(\psi_{1}+\psi_{2 })\gamma}}d\gamma\bigg{\}}\bigg{]}.\end{split} \tag{12}\]  
Now, using integral relationships, [47] (Equation (A.6)) and [46] (Equation (3.381.3)), we obtained \(\overline{C}_{OPRA}\) as:  
\[\begin{split}\overline{C}_{OPRA}=&\sum_{i=0}^{ \infty}\sum_{j=0}^{\infty}\sum_{n=0}^{N_{1}\mu_{1}+i-1}\sum_{l=0}^{N_{2}\mu_{2 }+j-1}\frac{\xi_{1}\xi_{2}\Delta\Pi\Psi B}{2\ln(2)(\psi_{1}+\psi_{2})^{n+l}} \bigg{[}\bigg{\{}\ln(\gamma_{o})\Gamma\Big{(}n+l+1,(\psi_{1}+\psi_{2})\gamma_ {o}\Big{)} \tag{13}\]  
where \(G_{\gamma}[\cdot]\) and \(\Gamma(\cdot,\cdot)\) are the Meijer G-function [46] (Sec. 9.3) and upper incomplete gamma function [46], respectively.