# Optimal Power Allocation with Hybrid Relaying Based on the Channel Condition
## 1 Introduction  
Cooperative communication has recently received much attention as a method to improve network performance [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11]. In multi-hop cooperative communications, the source transmits signal to relays that forward signals to the destination or other relays. There are two relaying strategies: amplify-and-forward (AF) and decode-and-forward (DF). In the AF relaying scheme, the relay simply amplifies the received signals from the source and retransmits them to the destination without performing any signal regeneration, which may lead to the propagation of noise and interference. For the DF relaying scheme, a relay decodes the received signals and retransmits the recovered signals to the destination. Although the DF relaying scheme achieves extra coding gain, the error propagation is caused by decoding errors in the relay. Two-way relay communication schemes are also drawing attention. Since the AF relaying scheme generally provides good spectral efficiency with much lower complexity than the DF relaying scheme, the authors of [10] consider the AF-based two-way relaying networks. In [10], the closed-form weightsfor distributed beamforming using zero-forcing are derived to cancel the inter-user interference in multi-user two-way AF relay networks. In [11], the performance of multiple AF relay networks is analyzed and investigated in consideration of significant practical issues such as time-selective fading due to node mobility and imperfect channel estimation. In addition, the generic expressions for the destination's signal-to-noise ratio (SNR), the system average bit error rate, the outage probability, and the capacity are derived in [11]. To obtain the advantages of both the AF and the DF strategies, hybrid relaying schemes are studied in [12; 13]. In [12], the authors analyze the bit error probability for both the AF relaying and the DF relaying with respect to SNR and propose a hybrid relaying scheme which changes the relaying scheme based on analyzed bit error probability. As in [12], the authors of [13] calculate the symbol error probability for both homogeneous relaying and hybrid relaying networks, and simulate the symbol error rate (SER) according to the location of the relay. The hybrid relaying schemes in [12; 13] have better bit error rate (BER) and SER performance than the simple homogeneous relaying schemes. These hybrid relaying networks obtain more gains than homogeneous relaying schemes. Recently, the power allocation problem in cooperative systems has been the focus of much research. In [14; 15], power allocation schemes are proposed to maximize the capacity under a sum transmit power constraint for AF relay and DF relay, respectively. Optimal power allocation schemes for hybrid networks are analyzed for a two-hop AF and DF cooperative relay system, employing the outage probability as the optimization criterion in [16]. In [17], optimal power allocation based on average end-to-end symbol error probability (SEP) as the optimization criterion is performed for a two-hop DF cooperative relay system. In [18; 19], the instantaneous received signal-to-noise ratio (SNR) and its approximate expression are exploited to obtain optimal power allocation for an AF multi-hop relaying system. The optimal power allocation based on outage probability in a DF multi-hop system is discussed in [20]. In [21], the power allocation scheme that minimizes a bit error rate (BER) at the destination for uncoded AF with a Rayleigh fading channel under sum transmit power consumption is proposed. The optimal power allocation strategy is proposed in [22] to maximize achievable secrecy rates under an overall transmit power constraint, assuming that a single relay is located at each individual hop. In hybrid relaying networks, the error performance is mainly affected by the received SNR, which is changed by the transmit power, channel power, and noise power. This paper proposes optimal power allocation schemes for hybrid relay networks within limited total power.