Cooperative Amplify-and-Forward Beamforming with Multiple Multi-Antenna Relays

In this paper, we consider beamforming (BF) for cooperative networks with one multi-antenna source, multiple multi-antenna amplify-and-forward (AF) relays, and one single-antenna destination. The source BF vector and the AF-BF matrices at the relays are optimized for maximization of the signal-to-noise ratio (SNR) at the destination under three different power constraints. In particular, we consider individual relay power constraints, a joint relay power constraint, and a joint power constraint for the source and the relays. We solve the associated optimization problems in two stages. In the first stage, we find the optimal AF-BF matrices for a given BF vector at the source. For the cases of individual and joint relay power constraints, closed-form solutions for the AF-BF matrices are provided, respectively. Furthermore, for the case of a joint source-relay power constraint, the AF-BF matrices are obtained in closed form up to a scalar power allocation factor and an efficient numerical algorithm for the power allocation between the source and the relays is given. In the second stage, the optimal source BF vectors are computed. Thereby, we show that for the joint relay and the joint source-relay power constraints, the resulting problem can be transformed into a non-convex polynomial programming problem which allows for an exact solution for small-scale networks. For large-scale networks and networks with individual relay power constraints, we propose efficient suboptimal optimization methods for the source BF vector. Simulation results show the benefits of having multiple antennas at the source and/or multiple multi-antenna relays and illustrate the performance differences introduced by the three different power constraints.