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In this letter, we consider a two-way relay channel (TWRC) with two end nodes and k relay nodes, where end nodes have the full channel-state information (CSI) and relay nodes only have the channel-amplitude information (CAI). With the objective of minimizing transmit power consumption at required end-to-end rates, energy-efficient relay selection (RS) and power allocation (PA) scheme is studied for TWRC based on analog network coding (ANC). Firstly, we propose an energy-efficient single RS and PA (E-SRS-PA) scheme, where the best relay node is selected to minimize total transmit power. Then, we prove that E-SRS-PA scheme is the optimal energy-efficient RS and PA (OE-RS-PA) scheme in ANC-based TWRC, and thus the optimal number of relay nodes to be selected in energy efficiency sense is equal to one. In addition, the closed-form expressions of optimal power allocation of E-SRS-PA scheme are derived. Numerical simulations confirm the optimality of proposed E-SRS-PA and demonstrate the energy efficiency of ANC-based TWRC compared with the other relaying schemes.
Zhou et al. (Wed,) studied this question.