| The relay channel is an information theoretic model for a communication system whereby a sender aims to communicate to a receiver with the help of a sender-receiver relay node. It represents the simplest model for a multi-hop wireless network and a full understanding of the limits of communication over such a channel can potentially shed light on the design of more efficient wireless networks. The capacity of the relay channel, however, has been established only for few special cases and little progress has been made toward solving the general case since the early 1980s.; In this dissertation, we present several new results on the capacity and minimum energy-per-bit for Additive White Gaussian Noise relay channels. We first review existing bounds on the capacity of the discrete memoryless relay channel and use them to derive bounds on the capacity of the AWGN relay channel. Motivated by practical wireless communication considerations, we introduce two different frequency-division AWGN relay channel models. Using known bounds, we are able to establish the capacity for the first model where the direct channel from the sender to the receiver and the channel from the sender to the relay use different frequency bands. For the second model, the direct channel from the sender to the receiver uses a different frequency band from that used in the channel from the relay to the receiver. In this case, the bounds are tight only under very restrictive conditions. We then investigate the practically interesting scenario where the relay can only send linear combinations of past received signals. Surprisingly, we find that this scheme can do almost as well as the much more sophisticated side information scheme, even outperforming it in some cases. Moreover, we establish the optimal achievable rates by this scheme for the second frequency-division relay channel model.; Finally we present the problem of minimum energy-per-bit for AWGN relay channels and find upper and lower bounds which differ by no more than a factor of 1.7 for the AWGN channel and 1.5 for the second frequency-division relay channel model. |
