Piecewise linear relaying: A low complexity parametric relaying

We investigate the Gaussian, three-nodes relay channel with orthogonal receive components, i.e., the transmitted signals from the source and the relay do not interfere with each other. We develop and analyze low-complexity symbol-wise (as opposed to block-wise) relaying strategies based on a one-dimensional, parametric piecewise linear (PL) mapping. We numerically compute and optimize the achievable rates obtained by the PL mapping for a certain number of design parameters. We conclude that optimized PL relaying always outperforms conventional instantaneous linear relaying (amplify-and-forward). We also illustrate that the proposed PL relaying can improve on sophisticated block Markov encoding (i.e., decode-and-forward) when the source-relay link is ill-conditioned (relative to other links). Furthermore, PL relaying can work at rates close to those achieved by side-information encoding (i.e., compress-and-forward), but at a much lower complexity.