Linear Feedback Coding for Gaussian Relay Channel With Various Feedback Links

Linear feedback coding scheme, such as the elegant Schalkwijk-Kailath (SK) scheme, receives much attention in the literature since its decoding error probability decreases as a second-order exponential in the coding blocklength. In recent years, a linear feedback scheme has been proposed for the Gaussian relay channel (GRC) with destination-source feedback, which combines the SK scheme and the amplify-and-forward (AF) relay strategy. Since there exists three possible feedback links in the GRC, then one question beckons: is there any rate gain if there exist multiple feedback links in the GRC, and can any other relay strategy outperform the AF strategy? In this paper, we answer this question by investigating four feedback models of the GRC, namely, the GRC with destination-source and destination-relay feedback, the GRC with destination-relay and relay-source feedback, the GRC with destination-source and relay-source feedback, and the GRC with all feedback links, respectively. We propose SK-type schemes for these feedback models, and numerical examples show that when the coding blocklength is not long, the rates of our proposed schemes almost approach their asymptotic values, and these rates may be larger than those of existing schemes in the literature. The study of this paper shows that different number/location of feedback links may bring additional rate gain in finite blocklength regime.