Bandwidth-Efficient Bit-Interleaved Coded Modulation Over NAF Relay Channels: Error Performance and Precoder Design

This paper investigates the error performance and precoder design for a bandwidth-efficient bit-interleaved coded modulation (BICM) system over a nonorthogonal amplify-and-forward (NAF) half-duplex single-relay channel. A tight union bound on the asymptotic bit error probability (BEP) is first derived for an arbitrary block length of 2N, which corresponds to the case of using a 2N × 2N precoder. This bound provides a useful tool for predicting the error performance. Attention is then paid to the coded NAF system that uses a 2 × 2 precoder, where a closed-form expression of the bound is obtained. Based on this expression, an optimal class of 2 × 2 precoders with respect to the asymptotic performance is then developed. Unlike the optimal precoders that are designed for uncoded systems, the derived precoder indicates that the source only needs to send the superposition of the first symbol and the rotated version of the second symbol in the first time slot while being silent in the remaining slot to achieve the best asymptotic performance. For good convergence property, it is further shown that a rotation angle that maximizes the minimum Euclidean distance of the superposition constellation should be used. An optimal rotation angle is then analytically determined for various modulation schemes. Both analytical and simulation results show that the proposed precoders not only exploit full cooperative diversity but offer a significant coding gain over the optimal precoders for uncoded NAF systems as well.