Precoding and Symbol Grouping for NAF Relaying in BICM Systems

This paper considers the precoder design for a bandwidth-efficient bit-interleaved coded modulation (BICM) system over nonorthogonal amplify-and-forward (NAF) single-relay channels with an arbitrary length of cooperative frame
$\hbox{2}N$
. Based on the tight union bound to the bit error probability (BEP), we first derive an asymptotic design criterion with regard to a general
$\hbox{2}N \times \hbox{2}N$
rotation matrix. This expression allows us to develop a class of precoders that not only achieves full cooperative diversity but optimizes the asymptotic error performance as well. Interestingly, the developed class of optimal precoders indicates that the source should keep silent in the cooperative phase. In the broadcasting phase, it is shown that power is equally distributed to
$\hbox{2}N$
information symbols at the source. By further examining the structure of the optimal class of
$\hbox{2}N \times \hbox{2}N$
precoders, it is then revealed that precoding over a group of at least two information symbols is sufficient to fully exploit diversity and coding advantages. Such a precoding technique, which is referred to as symbol grouping, therefore significantly reduces the system complexity without sacrificing the error performance. Among the pool of optimal precoders with respect to the asymptotic performance, the design of a precoder that provides a good convergence property is also discussed. Numerical examples show that the proposed precoder greatly improves the error performance of the cooperative system with low complexity.