## 5 Numerical Results and Discussions  
In \(\kappa-\mu\) shadowed fading environments, we evaluated the system performance of a dual-hop DF relaying system with beamforming using adaptive transmission techniques. The obtained analytical results (i.e., Equations (3), (13), (16), (19) and (22)) closely match with the results from the Monte Carlo simulations (where, each simulation point is average of \(2\times 10^{8}\) runs). Moreover, the expressions in analytic form are presented by an infinite expression of the form, which converges rapidly with several terms and are evaluated numerically. The infinite summations are truncated up to three decimal places and we used the numerical computation to ensure correctness. Figure 2 shows the outage performance with respect to the average SNR per hop (\(\overline{\gamma}=\overline{\gamma}_{1}=\overline{\gamma}_{2}\)) for various antenna numbers (\(N_{1}\) and \(N_{2}\)) and for different parameter values of \(\mu\) (\(\mu_{1}\), \(\mu_{2}\)). Figure 2 confirms that the numerical results obtained from Equation (5) are perfectly consistent with the simulation results. The fading parameter \(\mu\) and the number of antennas have a positive relationship with the system performance. Therefore, as expected, the system performance was improved with the fading parameter \(\mu\) and the number of antennas. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**First Hop/Second Hop** & \(\kappa_{1}\) & \(\mu_{1}\) & \(\kappa_{2}\) & \(\mu_{2}\) & \(m_{1}\) & \(m_{2}\) \\ \hline [PERSON]-\(\hat{m}\)/[PERSON]-\(\hat{m}\) & 0 & \(\hat{m}_{1}\) & 0 & \(\hat{m}_{2}\) & \(\infty\) & \(\infty\) \\ [PERSON]-\(\hat{m}\)/\(\kappa-\mu\) shadowed & 0 & \(\hat{m}_{1}\) & \(\kappa_{2}\) & \(\mu_{2}\) & \(\infty\) & \(m_{2}\) \\ [PERSON]-\(\hat{m}\)/\(\kappa-\mu\) & 0 & \(\hat{m}_{1}\) & \(\kappa_{2}\) & \(\mu_{2}\) & \(\infty\) & \(\infty\) \\ [PERSON]-\(\hat{m}\)/Rician-shadowed & 0 & \(\hat{m}_{1}\) & \(K_{2}\) & 1 & \(\infty\) & \(m_{2}\) \\ [PERSON]-\(\hat{m}\)/Rician & 0 & \(\hat{m}_{1}\) & \(K_{2}\) & 1 & \(\infty\) & \(\infty\) \\ [PERSON]-\(\hat{m}\)/Rayleigh & 0 & \(\hat{m}_{1}\) & 0 & 1 & \(\infty\) & \(\infty\) \\ Rayleigh/Rayleigh & 0 & 1 & 0 & 1 & \(\infty\) & \(\infty\) \\ Rayleigh/\(\kappa-\mu\) shadowed & 0 & 1 & \(\kappa_{2}\) & \(\mu_{2}\) & \(\infty\) & \(m_{2}\) \\ Rayleigh/\(\kappa-\mu\) & 0 & 1 & \(\kappa_{2}\) & \(\mu_{2}\) & \(\infty\) & \(\infty\) \\ Rayleigh/Rician-shadowed & 0 & 1 & \(K_{2}\) & 1 & \(\infty\) & \(m_{2}\) \\ Rayleigh/Rician & 0 & 1 & \(K_{2}\) & 1 & \(\infty\) & \(\infty\) \\ Rayleigh/Nakagami-\(\hat{m}\) & 0 & 1 & 0 & \(\hat{m}_{2}\) & \(\infty\) & \(\infty\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._In Figure 3, the average channel capacity is presented against the average SNR per hop using four adaptive transmission approaches, such as ORA, OPRA, CIFR, and TIFR. It can be observed that the analytical and the simulation results are closely matching with each other. Generally, the ORA technique performs well, as compared to the other techniques. In a high SNR regime, the ORA and OPRA techniques have an equivalent capacity performance. To improve the quality of links, the OPRA policy uses more transmit power, whereas the ORA policy only adapts its rate. Therefore, the ORA technique has a lower complexity, as compared to the OPRA technique. The CIFR technique performs poorly in the low and medium average SNR regions, when compared to the other techniques. However, in a high SNR regime, it attains the same capacity performance as TIFR. Figure 3: Average channel capacity for four different adaptive transmission methods with respect to the average SNR per hop when \(m_{1}=m_{2}=0.5\), \(\mu_{1}=\mu_{2}=1\), \(\kappa_{1}=\kappa_{2}=1\), and \(N_{1}=N_{2}=2\). Figure 2: Outage probability against average SNR per hop for different arrangements of antenna and fading parameter \(\mu\) when \(\kappa_{1}=\kappa_{2}=1\), \(m_{1}=m_{2}=2\), and \(\gamma_{th}=10\) dB.