# Optimal Power Allocation with Hybrid Relaying Based on the Channel Condition
## 4 Simulation Results
We show and compare the values of achievable rates for the DF-AF relay system in Tables 4-6. The \(\alpha_{da}\) decreases for Case a(1) and increases for Case b(1) as \(P\) increases. As mentioned in Case a(2), \(\alpha_{da}\) is \(1/2\) for \(\frac{P_{s}|h_{2}|^{2}}{P|h_{2}|^{2}}\geq 1/2\) and \(\frac{P_{s}|h_{2}|^{2}}{P|h_{2}|^{2}}\) for \(\frac{P_{s}|h_{2}|^{2}}{P|h_{2}|^{2}}\leq 1/2\). In other words, \(\alpha_{da}\) is \(1/2\) for \(P\leq\frac{2P_{s}|h_{2}|^{2}}{|h_{2}|^{2}}\) and \(\frac{P_{s}|h_{2}|^{2}}{P|h_{2}|^{2}}\) for \(P\geq\frac{2P_{s}|h_{2}|^{2}}{|h_{2}|^{2}}\). When \(P_{s}=10\) dB and \(|h_{1}|^{2}=|h_{2}|^{2}=0.2\), \(\frac{2P_{s}|h_{2}|^{2}}{|h_{2}|^{2}}\) is \(13.0103\) dB. Therefore, as shown in Figure 6b, \(\alpha_{da}\) for Case a(2) is \(1/2\) for \(P\leq 13.0103\) dB and \(10^{(1-0.1P)}\) for \(P\geq 13.0103\) dB. The achievable rate of the optimal power allocation scheme for Case a(2) is the same as for the grid search schemes and the equal power allocation scheme when \(P\leq 13.0103\) dB. Then, the optimal power allocation scheme for Case a(2) provides a higher achievable rate than the grid search schemes and the equal power allocation scheme when \(P\geq 13.0103\) dB. Figure 5: Performance for the AF-DF relay network when \(P_{s}=10\) dB. (**a**) Achievable rates for the power allocation schemes; (**b**) \(\alpha_{ad}\) for the optimal power allocation scheme and grid search schemes. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(P\) & **Proposed Allocation** & **Grid Search with \(\Delta=0.01\)** & **Grid Search with \(\Delta=0.1\)** & **Equal Allocation** \\ \hline
3 dB & 0.0890 & 0.0889 & 0.0839 & 0.0670 \\
6 dB & 0.1654 & 0.1645 & 0.1628 & 0.1250 \\
12 dB & 0.4611 & 0.4604 & 0.4402 & 0.3604 \\
15 dB & 0.6461 & 0.6458 & 0.6401 & 0.5276 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Achievable rates for the AF-DF relay network when \(|h_{1}|^{2}=|h_{2}|^{2}=|h_{3}|^{2}=0.1\). Figure 7a,b show the achievable rates of the proposed power allocation schemes for hybrid three-hop relay networks when the SNR of the first hop \(\gamma_{s1}\) is \(0\) dB, \(-3\) dB, and \(-6\) dB. When \(|h_{3}|^{2}\) has a greater value than \(|h_{2}|^{2}\), it is observed that the achievable rate for DF-AF relay network is greater than that of AF-DF relay networks regardless of \(\gamma_{s1}\). On the other hand, the achievable rate for the AF-DF relay network is greater than for DF-AF relay networks when \(|h_{2}|^{2}\) has a greater value than \(|h_{3}|^{2}\). As mentioned, the SNR at destination \(\gamma_{ad}\) for the AF-DF relay network is determined as the minimum value between the SNR at the second relay \(\gamma_{ad,s2}\) and the SNR of the third hop \(\gamma_{ad,2d}\). Because \(\gamma_{ad,s2}\) has a form similar to the harmonic mean between \(\gamma_{s1}\) and the SNR of the second hop, the increment of \(\gamma_{ad,s2}\) is less than that of \(\gamma_{ad,2d}\) when the increments in SNR of each hop are the same. Therefore, to maximize \(\gamma_{ad}\), we need to further increase \(\gamma_{ad,s2}\), which does not increase as much as \(\gamma_{ad,2d}\). For a given \(\gamma_{s1}\), \(\gamma_{ad,s2}\) can be increased by increasing \(|h_{2}|^{2}\). Unlike the AF-DF relay network, the SNR at \begin{table}
\begin{tabular}{c c c c} \hline \(P\) & **Proposed Allocation** & **Grid Search with \(\Delta=0.01\)** & **Grid Search with \(\Delta=0.1\)** & **Equal Allocation** \\ \hline
3 dB & 0.1845 & 0.1845 & 0.1827 & 0.1694 \\
6 dB & 0.4196 & 0.4195 & 0.4194 & 0.3725 \\
12 dB & 1.4160 & 1.4122 & 1.3892 & 1.2404 \\
15 dB & 2.1634 & 2.1588 & 2.0961 & 1.7872 \\ \hline \end{tabular}
\end{table} 
Table 6: Achievable rates for the DF-AF relay network in Case b(1): \(|h_{1}|^{2}=|h_{2}|^{2}=0.2\) and \(|h_{3}|^{2}=2\). Figure 6: Performance for the DF-AF relay network when \(P_{3}=10\) dB. (**a**) Achievable rates for the power allocation schemes; (**b**) \(\alpha_{da}\) for the optimal power allocation scheme and grid search schemes.