Optimal Precoder for Amplify-and-Forward Half-Duplex Cooperative System

In this paper, we design an optimal precoder for the amplify-and-forward (AF) half-duplex cooperative system. We first present asymptotic pairwise error probability (PEP) expression and identify the corresponding terms for diversity and coding gains. By employing the properties of the Farey sequence in number theory, we obtain a closed form optimal precoder to achieve both full diversity as well as maximum coding gain for square quadrature amplitude modulation (QAM) signals. Simulation results indicate that the proposed design significantly improves the BER performance of the relay system.