Bounds on capacity and minimum energy-per-bit for AWGN relay channels

Upper and lower bounds on the capacity and minimum energy-per-bit for general additive white Gaussian noise (AWGN) and frequency-division AWGN (FD-AWGN) relay channel models are established. First, the max-flow min-cut bound and the generalized block-Markov coding scheme are used to derive upper and lower bounds on capacity. These bounds are never tight for the general AWGN model and are tight only under certain conditions for the FD-AWGN model. Two coding schemes that do not require the relay to decode any part of the message are then investigated. First, it is shown that the "side-information coding scheme" can outperform the block-Markov coding scheme. It is also shown that the achievable rate of the side-information coding scheme can be improved via time sharing. In the second scheme, the relaying functions are restricted to be linear. The problem is reduced to a "single-letter" nonconvex optimization problem for the FD-AWGN model. The paper also establishes a relationship between the minimum energy-per-bit and capacity of the AWGN relay channel. This relationship together with the lower and upper bounds on capacity are used to establish corresponding lower and upper bounds on the minimum energy-per-bit that do not differ by more than a factor of 1.45 for the FD-AWGN relay channel model and 1.7 for the general AWGN model.

I. Introduction

THE relay channel, first introduced by van der Meulen [1] in 1971, consists of a sender–receiver pair whose communication is aided by a relay node. In [1] and [2], simple lower and upper bounds on the capacity of the discrete-memoryless relay channel were established. In [3] and [4], capacity theorems were established for: i) physically degraded and reversely degraded discrete memoryless relay channels, ii) physically degraded and reversely degraded additive white Gaussian noise (AWGN) relay channels with average power constraints, iii) deterministic relay channels, and iv) relay channels with feedback. A max-flow min-cut upper bound and a general lower bound based on combining the generalized block-Markov and side-information coding schemes were also established in [4]. In [5], the capacity of the relay channel with one deterministic component was established. It is interesting to note that in all special cases where the relay channel capacity is known, it is equal to the max-flow min-cut bound. Generalizations of some of the single-relay channel results to channels with many relays were given in [6]. In [7], Aref established the capacity for a cascade of degraded relay channels. The relay channel did not receive much attention and no further progress was made toward establishing its capacity for a long time after this early work.