Optimal Fixed Gain Linear Processing for Amplify-and-Forward Multichannel Relaying

In this correspondence, we consider the problem of linear processing design at the relay for amplified-and-forward relaying in a multichannel system. Assuming a fixed-gain power amplification at the relay, we study the linear processing structure to maximize the end-to-end achievable rate. For both the cases of relaying with or without direct path, we show that the optimal unitary processing matrix is of permutation structure, i.e., channel pairing is optimal. Furthermore, in each case, the explicit optimal channel pairing strategy is obtained based on sorting certain function of received SNR over the incoming and outgoing subchannels. This result is especially noticeable for the case with direct path, where the optimal linear processing was not known before under any power allocation. Specifically, we show that the pairing is according to the ordering of the relative SNR ratio on a subchannel over first hop to its direct path, and that of SNR strengths on subchannels over the second hop. Simulation results are presented to demonstrate the achievable gain of optimal channel pairing over non-optimal linear processing or no-pairing cases. It is also shown that the performance of channel pairing under the simple fixed-gain power allocation outperforms that under the traditional uniform power allocation.