Two-Way Relaying With Multi-Antenna Sources: Beamforming and Antenna Selection

We propose and analyze two multiple-input-multiple-output (MIMO) two-way relaying schemes with an amplify-and-forward protocol in Nakagami-m fading channels, where multi-antenna sources communicate via a single-antenna relay. Specifically, we present a new framework for the comparative analysis of beamforming and antenna selection with nonidentical fading parameter m in the two source-relay links. To facilitate the comparison, we derive new exact, approximate, and asymptotic expressions for the sum symbol error rate (SSER) with M-ary phase-shift keying (M-PSK) and M-ary quadrature amplitude modulation (M-QAM). Based on the asymptotic SSER, we prove that beamforming and antenna selection have the same diversity order. The diversity order is dominated by the weaker source-relay link, which is determined by the product of the number of source antennas and the fading parameter. We proceed to characterize the fundamental difference between the two schemes in terms of their array gains and average signal-to-noise ratios (SNRs). To obtain further insights, we address the key question of "How to allocate the total transmit power such that the SSER is minimized?" Our answer is given in the form of new concise expressions for the power-allocation factor that optimally distributes the total transmit power between the sources and the relay. A pivotal conclusion is reached that antenna selection offers the same SSER as beamforming when the source in the weaker link is equipped with a single antenna.