This paper considers the Gaussian half-duplex relay channel (G-HD-RC): a channel model where a source transmits a message to a destination with the help of a relay that cannot transmit and receive at the same time. It is shown that the cut-set upper bound on the capacity can be achieved to within a constant gap, regardless of the actual value of the channel parameters, by either partial-decode-and-forward or compress-and-forward. The performance of these coding strategies is evaluated with both random and deterministic switch at the relay. Numerical evaluations show that the actual gap is less than what analytically obtained, and that random switch achieves higher rates than deterministic switch. As a result of this analysis, the generalized degrees-of-freedom of the G-HD-RC is exactly characterized for this channel. In order to get insights into practical schemes for the G-HD-RC that are less complex than partial-decode-and-forward or compress-and-forward, the exact capacity of the linear deterministic approximation (LDA) of the G-HD-RC at high signal-to-noise-ratio is determined. It is shown that random switch and correlated nonuniform inputs bits are optimal for the LDA. It is then demonstrated that deterministic switch is to within one bit from the capacity. This latter scheme is translated into a coding strategy for the original G-HD-RC and its optimality to within a constant gap is proved. The gap attained by this scheme is larger than that of partial-decode-and-forward, thereby pointing to an interesting practical tradeoff between gap to capacity and complexity.
In @cite_8 , Kramer shows that larger rates can be achieved by using a random transmit listen switch strategy at the relay. In this way, the source and the relay can harness the randomness that lies in the switch in order to transmit extra information. An important observation of @cite_8 is that there is no need to develop a separate theory for memoryless networks with HD nodes as the HD constraints can be incorporated into the memoryless FD framework. In this work we shall adopt this approach in deriving outer and inner bounds for HD relay networks.
@cite_8
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Relay channels where terminals cannot receive and transmit at the same time are modeled as being memoryless with cost constraints. Cost functions are considered that measure the power consumed in each of three sleep-listen-or-talk (SLoT) modes, as well as the fraction of time the modes are used. It is shown that strategies that have the SLoT modes known ahead of time by all terminals are generally suboptimal. It is further shown that Gaussian input distributions are generally suboptimal for Gaussian channels. For several types of models and SLoT constraints, it is shown that multi-hopping (or decode-andforward) achieves the information-theoretic capacity if the relay is geometrically near the source terminal, and if the fraction of time the relay listens to the source is lower bounded by a positive number. SLoT constraints for which the capacity claim might not be valid are discussed. Finally, it is pointed out that a lack of symbol synchronization between the relays has little or no effect on the capacity theorems if the signals are bandlimited and if independent input signals are optimal.