Compress-and-Forward Performance in Low-SNR Relay Channels

In this paper, we study the Gaussian relay channels in the low signal-to-noise ratio (SNR) regime with the time-sharing compress-and-forward (CF) scheme, where at each time slot all the nodes keep silent at the first fraction of time and then transmit with CF at a higher peak power in the second fraction. Such a silent vs. active two-phase relay scheme is preferable in the low-SNR regime. With this setup, the upper and lower bounds on the minimum energy per bit required over the relay channel are established under both full-duplex and half-duplex relaying modes. In particular, the lower bound is derived by applying the max-flow min-cut capacity theorem; the upper bound is established with the aforementioned time-sharing CF scheme, and is further minimized by letting the active phase fraction decrease to zero at the same rate as the SNR value. Numerical results are presented to validate the theoretical results.