General-Rank Beamforming for Multi-Antenna Relaying Schemes

We consider a wireless network consisting of a single-antenna transmitter, a single-antenna receiver, and a multi-antenna relay node. For such a network, we introduce the novel approach of general rank beamforming. In this approach, the relay multiplies the vector of its received signals by a general-rank complex matrix to obtain a new vector. Each entry of this new vector is then transmitted on one of the antennas available at the relay. We show that maximizing the receiver SNR subject to total relay power constraint yields a closed. form solution for the beamforming matrix. We also prove that if the channel coefficients from the transmitter to the relay antennas and those from the relay antennas to the receiver are statistically independent, the general rank beamforming approach results in a rank-one solution for the beamforming matrix.