Optimal Precoder and Symbol Grouping for Bandwidth-Efficient Bit-Interleaved Coded Modulation over NAF Single-Relay Channels

This paper considers the precoder design for a bandwidth-efficient bit-interleaved coded modulation (BICM) over non-orthogonal amplify-and-forward (NAF) single-relay channels with an arbitrary length of cooperative frame 2N. Based on the tight union bound on the bit error probability (BEP), we first derive an asymptotic design criterion with regard to a general 2N × 2N rotation matrix. This expression allows us to develop a class of precoder that not only achieves full cooperative diversity but also optimizes the asymptotic error performance. Interestingly, the developed class of optimal precoder indicates that the source should be kept silent in the cooperative phase. In the broadcasting phase, it is shown that power is distributed equally to 2N information symbols at the source. By further examining the structure of the optimal class of 2N × 2N precoders, we then reveal that precoding over a group of at least 2 information symbols is sufficient to fully exploit diversity and coding advantages. Such precoding technique, which is referred to as symbol grouping, therefore significantly reduces the system complexity without degrading the error performance.