# Optimal Power Allocation with Hybrid Relaying Based on the Channel Condition
## 2 System Model  
In [23], the comparisons of throughput performance for multi-hop strategies are shown by computational simulation results. The throughput performance for systems with two-hop and four-hop relaying is poorer than for systems with three-hop relaying. The reasons are summarized as follows. First, it is difficult to find the node with the best transmission rate for the two-hop relaying system because of the short transmission range of the ad hoc interface. Also, in systems with more than three hops, there is a greater overhead for the routing or forwarding delay. The three-hop relaying system can greatly facilitate in the inter-cell communication because the node has a high probability of reaching a neighboring base station. Therefore, the number of hops in this paper's relay network is confined to three. Figure 1 shows the system model consisting of a source \(s\), the first relay \(r_{1}\), the second relay \(r_{2}\), and a destination \(d\). The nodes operate in the half-duplex mode, i.e., they are not able to receive and transmit at the same time and same frequency. We assume that the transmit time from one node to another node is the same. In addition, we assume that there is no delay for processing the signal in AF and DF relays. In hybrid three-hop relay systems, the transmitted signal from source can be received at destination after processing the signals at two relays. If the first relay amplifies-and-forwards the received signal and the second relay decodes-and-forwards the received signal, the system model is considered as the AF-DF relay system. On the other hand, if the first relay decodes-and-forwards the received signal and the second relay amplifies-and-forwards the received signal, the system model is  
considered as a DF-AF relay system. The AF-DF and DF-AF relay systems have distinct error rates and achievable throughput in accordance with the different channel conditions between two nodes. We assume that the channel gains are acquired from channel state information (CSI) of a system by using a reference signal (RS) of the 3 rd Generation Partnership Project (3 GPP) Long Term Evolution (LTE) and all relays in the system are aware of the used power for the transmission of a signal. In the case of three-hop relay network, three RSs should be allocated and the destination provide feedback on the CSI through reverse links of the relay network. For channel estimation, various schemes can be considered [27; 28]. However, a perfect CSI is assumed to compare the maximum performance with other power allocation schemes. In addition, it is assumed that the total power for relaying is fixed. Since the relays also need power to transmit their own signal, this condition is necessary. In the case of an equal power allocation scheme, the fixed total power is equally allocated to all relays. However, since channel condition is not of concern, the equal power allocation scheme inefficiently allocates the power resource between relays. Therefore, the optimization procedures should be applied for system power efficiency. This fixed total power can be normalized for comparison with other power allocation schemes [24; 25]. The purpose of hybrid relay network is to achieve both high throughput performance and simple implementation. The different properties in AF-DF and DF-AF relay networks result in these two schemes showing different performance with respect to achievable throughput and implementation, in accordance with channel conditions. Generally, because the AF scheme severely amplifies the noise power, the error performance for the AF scheme is poorer than that of the DF scheme. The error performance of the AF scheme is almost the same as that for the DF scheme when the channel condition is good. However, because the DF scheme always knows the channel state information (CSI) needed to decode the received signals, one of main disadvantages for the DF scheme is that the real-time implementation is more difficult than in the AF scheme for the multi-hop transmission system.