Amplify-and-Forward Multi-Antenna Beamforming with Joint Source-Relay Power Constraint

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 at the destination under a joint power constraint for the source and the relays. We solve the associated optimization problem in two stages. In the first stage, we find the optimal AF-BF matrices for a given BF vector at the source. In particular, the direction of the AF-BF matrices is derived in closed form and an efficient numerical algorithm for the convex power allocation problem between the source and the relays is provided. In the second stage, the optimal source BF vectors are computed. Thereby, we show that 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, we propose efficient suboptimal optimization methods for the source BF vector.