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

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.Comment: 28 page

On the Reliability Function of the Discrete Memoryless Relay Channel

Bounds on the reliability function for the discrete memoryless relay channel are derived using the method of types. Two achievable error exponents are derived based on partial decode-forward and compress-forward which are well-known superposition block-Markov coding schemes. The derivations require combinations of the techniques involved in the proofs of Csisz\'ar-K\"orner-Marton's packing lemma for the error exponent of channel coding and Marton's type covering lemma for the error exponent of source coding with a fidelity criterion. The decode-forward error exponent is evaluated on Sato's relay channel. From this example, it is noted that to obtain the fastest possible decay in the error probability for a fixed effective coding rate, one ought to optimize the number of blocks in the block-Markov coding scheme assuming the blocklength within each block is large. An upper bound on the reliability function is also derived using ideas from Haroutunian's lower bound on the error probability for point-to-point channel coding with feedback.Comment: To appear in the IEEE Transactions on Information Theory; Presented in part at the 2013 ISI