On the optimality of successive decoding in compress-and-forward relay schemes

In the classical compress-and-forward relay scheme developed by (Cover and El Gamal, 1979), the decoding process operates in a successive way: the destination first decodes the compressed observation of the relay, and then decodes the original message of the source. Recently, two modified compressand- forward relay schemes were proposed, and in both of them, the destination jointly decodes the compressed observation of the relay and the original message, instead of successively. Such a modification on the decoding process was motivated by realizing that it is generally easier to decode the compressed observation jointly with the original message, and more importantly, the original message can be decoded even without completely decoding the compressed observation. Thus, joint decoding provides more freedom in choosing the compression rate at the relay, i.e., the relay's observation can be compressed at a rate higher than supportable by successive decoding. However, the question remains whether this freedom of choosing a higher compression rate at the relay improves the achievable rate of the original message. It has been shown in (El Gamal and Kim, 2010) that the answer is negative in the single relay case, and the achievable rate obtained in (Cover and El Gamal, 1979) with successive decoding is still the best. In this paper, we further demonstrate that in the case of multiple relays, there is no improvement on the achievable rate by joint decoding either. More interestingly, it is discovered that any compression rates higher than supportable by successive decoding will actually result in a strictly lower achievable rate for the original message. Therefore, to maximize the achievable rate for the original message, the compression rates should always be chosen to be supportable by successive decoding. The freedom of choosing higher compression rates introduced by joint decoding is actually obtained at the sacrifice of the achievable rate for the original message. This phenomenon is also shown to exist under the repetitive encoding framework recently proposed by (Lim, Kim, El Gamal, and Chung, 2010).