Power allocation in relay channels under a global power constraint using virtual nodes

Relay channels have been extensively studied in the literature since the seminal paper by Cover and El Gamal. Nevertheless, characterizing the capacity of relay channels still presents open issues. While numerous works addressed this problem with constant powers or targeted the sum-rate optimization, computing the capacity in the case of a global power constraint was less studied. In this paper, we introduce the concept of virtual nodes to derive analytical expressions of the relay channel capacity as a function of the total power. This transformation leads to simple closed-form expressions of the upper bound and decode-and-forward (DF) lower bound on the capacity of the full- and half-duplex relay channels. The half-duplex study is separated into low and high signal-to-noise ratio (SNR) cases. The impact of these approximations is evaluated and found to achieve a large part of the maximal capacity in the worst case where the equivalent received SNR is neither low nor high, typically between 0-10dB.