Capacity-Approaching Design for Half-Duplex NAF Relay Channels

In this paper, we design a capacity-approaching coded modulation scheme for half-duplex non-orthogonal amplify-and-forward (NAF) single-relay channel. We apply the idea of multi-dimensional (multi-D) mapping employed in precoded multiple cooperative frames, concatenated with a simple outer binary code. Using union bounding techniques, it is first shown that for any unitary and full-diversity rotation G, the optimal multi-D labeling for NAF relaying shall maximize the average Euclidean distance between all pairs in the multi-D rotated constellation whose labels differ in only one bit. The extrinsic information transfer (EXIT) charts are then used to match the outer code, the multi-D mapping, and the precoder for near-capacity performance. It is demonstrated that the proposed scheme is promising for NAF relaying, in the sense that it can operate below the achievable rate obtained by using a conventional modulation scheme, i.e., the constrained capacity. In particular, for various spectral efficiencies, we obtain the bit error rate (BER) of 10^-5 or lower at a signal-to-noise ratio (SNR) that is 0.45dB-1dB below the traditional achievable rate and within 1.55dB-1.95dB from the ergodic capacity with Gaussian inputs.