On Instantaneous Relaying

The Gaussian, three-node relay channel with orthogonal receive components (i.e., the transmitted signals from the source and the relay do not interfere with each other) is investigated. For such channels, linear relaying is a suboptimal strategy in general. This is because a linear scheme merely repeats the received noisy signal and does not utilize the available degrees of freedom efficiently. At this background, nonlinear, symbol-wise (as opposed to block-wise) relaying strategies are developed to compensate for the shortcomings of the linear strategy. Optimal strategies are presented for two special cases of the general scenario, and it is shown that memoryless relaying can achieve the capacity. Furthermore, for the general Gaussian relay channel, a parametric piecewise linear (PL) mapping is proposed and analyzed. The achievable rates obtained by the PL mapping are computed numerically and optimized for a certain number of design parameters. It is concluded that optimized PL relaying always outperforms conventional instantaneous linear relaying (amplify-and-forward). It is also illustrated that the proposed PL relaying scheme 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.