Achievable Rates for Gaussian Degraded Relay Channels with Non-Vanishing Error Probabilities

TL;DR

This paper characterizes the psilon-capacity of Gaussian degraded relay channels with expected power constraints, showing it exceeds the classical capacity for psilon>0 and analyzing the second-order coding rate terms.

Contribution

It provides the first characterization of psilon-capacity for Gaussian degraded relay channels with expected power constraints, including achievability, converse, and second-order analysis.

Findings

psilon-capacity is strictly larger than capacity for psilon>0

Achievability uses block Markov coding and power control techniques

Converse bounds are derived from hypothesis testing and conditioning arguments

Abstract

This paper revisits the Gaussian degraded relay channel, where the link that carries information from the source to the destination is a physically degraded version of the link that carries information from the source to the relay. The source and the relay are subject to expected power constraints. The $\varepsilon$-capacity of the channel is characterized and it is strictly larger than the capacity for $\varepsilon>0$, which implies that the channel does not possess the strong converse property. The proof of the achievability part is based on several key ideas: block Markov coding which is used in the classical decode-forward strategy, power control for Gaussian channels under expected power constraints, and a careful scaling between the block size and the total number of block uses. The converse part is proved by first establishing two non-asymptotic lower bounds on the error probability, which are derived from the type-II errors of some binary hypothesis tests. Subsequently, each lower bound is simplified by conditioning on an event related to the power of some linear combination of the codewords transmitted by the source and the relay. Lower and upper bounds on the second-order term of the optimal coding rate in terms of blocklength and error probability are also obtained.