# Optimal Power Allocation with Hybrid Relaying Based on the Channel Condition
## 1 Introduction
The proposed schemes provide a higher achievable rate than the equal power allocation scheme and can approach the maximum achievable rate with lower power than an equal power allocation scheme. Also, after applying the proposed power allocation schemes, the achievable rates for two hybrid relay networks, i.e., AF-DF and DF-AF, are distinct in accordance with the differences in channel conditions from one relay to another and from the relay to a destination. In other words, the proposed scheme optimizes hybrid relaying of two relay links by selecting a better relaying scheme with optimal power allocation. Optimization for two relay links provides better reliability than optimization of only one relay link. In inter-cell communication systems, which are more common than intra-cell communication systems, three-hop relaying transmission is sufficient to achieve optimal throughput and to find the optimal relay node [23]. The simulation results in [23] show that three-hop relaying transmission has better throughput performance than two-hop and four-hop relaying transmission. For transmission with the two-hop relaying network, the transmission range is short and this causes a decrease in achievable throughput in the inter-cell communication. Due to the short range, it is more difficult for the two-hop relaying system to select a better relaying node in terms of throughput performance than the three-hop relaying system. Also, for transmission with four or more relaying hops, the throughput can be severely decreased because the routing with four or more relaying hops increases the forwarding delay and causes a greater overhead and signal processing delay for the overall system. Therefore, the number of hops in this paper is confined to three. If more nodes than three are neededdue to high path loss, the problem becomes the optimization of general N-hop relaying. If N is larger than three, optimization of general N-hop relaying is inefficient. To avoid the inefficient optimization, the N-hop relaying can be treated as successive optimization of two relay links with the optimization of one relay link. In [24; 25], to increase the channel capacity, cooperative communication systems with high diversity gain and the relay selection are used in consideration of the power allocation. However, because cooperative communication generally has a complicated structure, the following problems occur, as pointed out in [26]. Firstly, a sophisticated scheduler is needed for a large number of nodes, which are used as relays to form complex cooperative communication. Also, an additional overhead is required for channel estimation, synchronization, and security for multiple nodes. However, the proposed schemes improve the channel capacity through optimal power allocation and a hybrid relaying network in accordance with channel conditions without a sophisticated scheduler and additional overhead. The proposed schemes enable the achievable rate to maximize by adaptively allocating the power to the first and the second relay nodes. For adaptive power allocation of each relay node, we derive the transmit power values in a closed form for each relay network according to the channel condition. Analytical solutions are derived, and the proposed power allocation schemes are compared with the equal power allocation scheme and the grid search schemes. In addition, we compare the achievable rates of the proposed power allocation schemes when the SNR of the first hop is the same. The simulation results show that the proposed optimal power allocation scheme requires lower transmit power to achieve a specific achievable rate than the equal power allocation scheme. Therefore, this paper contributes to a reduction in the lower limit of transmit power consumption to satisfy the achievable rate in cooperative communication. In terms of next-generation systems, green communication has become popular.