# Optimal Power Allocation with Hybrid Relaying Based on the Channel Condition
## 4 Simulation Results  
This section presents the achievable rates of the proposed power allocation scheme, grid search schemes with step sizes \(\Delta=0.01\) and 0.1, and the equal power allocation scheme for hybrid three-hop relay networks. For the equal power allocation schemes, \(\alpha_{ad}\) and \(\alpha_{da}\) are fixed to \(1/2\). Also, we assume that any delay components made in relays are removed perfectly and are ignored. Figure 5a,b shows the achievable rates and \(\alpha_{ad}\) for the AF-DF relay network when \(P_{s}=10\) dB. From Figure 5a, it is observed that the achievable rate of the optimal power allocation scheme is greater than for the grid search schemes and the equal power allocation scheme regardless of channel gains. The grid search scheme with step size \(\Delta=0.01\) shows the performance close to the optimal power allocation scheme because this scheme can precisely find \(\alpha_{ad}\) in 0.01 unit which maximizes the achievable rate. Since the grid search scheme with \(\Delta=0.1\) can find \(\alpha_{ad}\) in only 0.1 units, it is more difficult to find the \(\alpha_{ad}\) that provides optimal performance. As shown in Figure 5b, the difference of \(\alpha_{ad}\) between the optimal power allocation and the grid search scheme becomes larger as the step size \(\Delta\) becomes larger. Therefore, the grid search scheme with \(\Delta=0.1\) shows poorer performance than that with \(\Delta=0.01\). In Tables 2 and 3, we show and compare the values of achievable rate for power allocation schemes when \(P=3,6,12,15\) dB. From Tables 2 and 3, the optimal power allocation scheme shows the best performance and the equal power allocation scheme shows the worst performance. In addition, the grid search scheme with \(\Delta=0.01\) shows a better performance than with \(\Delta=0.1\). Among the results in Figure 5a, the achievable rates expressed by the dashed line for optimal power allocation and equal power allocation converge after the power constraint of 10 dB. This can be understood from Equations (17) and (18). After the power constraint of 10 dB, the \(\gamma_{ad}\) of the two allocation schemes is determined by Equation (17) and is hardly subject to \(\alpha_{ad}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Hybrid Relay Network** & **Proposed Allocation** & **Grid Search with \(\Delta=0.01\)** & **Grid Search with \(\Delta=0.1\)** \\ \hline \(\alpha_{ad}\) for the AF-DF relay network & \(14O^{2}+6O\) & \((11O^{2}+3O)*99+98O\) & \((11O^{2}+3O)*9+8O\) \\ \(\alpha_{da}\) for the DF-AF relay network & \(9O^{2}+10O\) & \((12O^{2}+6O)*99+98O\) & \((12O^{2}+6O)*9+8O\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Complexity of proposed and grid search schemes to find \(\alpha_{ad}\) and \(\alpha_{da}\). From Figure 5b, we can know that \(\alpha_{ad}\) is greater than \(1/2\) when \(|h_{1}|^{2}=|h_{2}|^{2}=|h_{3}|^{2}=0.1\). As mentioned, the SNR at destination \(\gamma_{ad}\) is determined by the minimum value between the SNR at the second relay \(\gamma_{ad,s2}\) and the SNR of the third hop \(\gamma_{ad,2d}\). In addition, the increment of \(\gamma_{ad,s2}\) is less than that of \(\gamma_{ad,2d}\) when increments of the transmit power from each relay are the same. To maximize \(\gamma_{ad}\), we need to further increase \(\gamma_{ad,s2}\) which does not increase as much as \(\gamma_{ad,2d}\). Therefore, we should allocate more power at the first relay than the second relay to increase \(\gamma_{ad,s2}\) more than \(\gamma_{ad,2d}\). Figure 6a,b show the achievable rates and \(\alpha_{da}\) for DF-AF relay network when \(P_{s}=10\) dB. Case a(1) and a(2) are described when \(|h_{2}|^{2}\
eq|h_{3}|^{2}\) and \(|h_{2}|^{2}=|h_{3}|^{2}\), respectively. Then, Case b(1) is described when \(|h_{2}|^{2}\
eq|h_{3}|^{2}\). From Figure 6a, it is observed that the optimal power allocation schemes for Case a(1) and b(1) provide a higher achievable rate than the grid search schemes and equal power allocation scheme.