Multiple-Frame Precoding for NAF Relaying Over Rayleigh Fading Channels

This paper proposes a precoding scheme over multiple cooperative frames to exploit time and cooperative diversities over a non-orthogonal amplify-and-forward (NAF) half-duplex single-relay Rayleigh fading channel. Focusing on a coded system under the framework of bit interleaved coded modulation (BICM), we first derive a union bound on the bit error probability (BEP) of the proposed system. It is shown that the diversity gain function is the (N_f ·d_H)th power of that of uncoded cooperative systems, where N_f is the number of precoded cooperative frames, and d_H is the minimum Hamming distance of the outer code. An optimal class of precoders is then derived to optimize the asymptotic coding gain. It is then shown that the source should transmit a superposition of all symbols in the broadcasting phases while being silent in all cooperative phases for the best asymptotic performance. By further analyzing the first iteration performance, a design criterion is then developed to find optimal superposition angles for good convergence behavior. A pragmatic approach is then proposed to find good rotation angles. Analytical and simulation results show that the proposed scheme provides a significantly higher order of time and cooperative diversities and better coding gains than previous precoding schemes.