Gaussian Relay Channel with Noiseless Feedback: Revisit

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 two feedback links in the GRC, and can any other relay strategy outperform the AF strategy? In this paper, we answer this question by investigating two feedback models of the GRC, namely, the GRC with destination-source and destination-relay feedback, and the GRC with destination-relay and relay-source feedback. We propose two SK-type schemes which are respectively based on the decode-and-forward (DF) relay strategy and the compress-and-forward (CF) relay strategy. Numerical examples show that the achievable rates of our proposed schemes may be larger than that of the AF relay strategy based SK-type scheme in the literature.