Semidefinite Relaxation and Approximation Analysis of a Beamformed Alamouti Scheme for Relay Beamforming Networks

In this paper, we study the amplify-and-forward (AF) schemes in two-hop one-way relay networks. In particular, we consider the multigroup multicast transmission between long-distance users. Given that perfect channel state information is perceived, our goal is to design the AF process so that the max-min-fair (MMF) signal-to-interference-plus-noise ratio (SINR) is optimized subject to generalized power constraints. We propose a rank-two beamformed Alamouti (BFA) AF scheme and formulate the corresponding AF design problem as a \emph{two-variable} fractional quadratically-constrained quadratic program (QCQP), which is further tackled by the semidefinite relaxation (SDR) technique. We analyze the approximation quality of two-variable fractional SDRs under the Gaussian randomization algorithm. These results are fundamentally new and reveal that the proposed BFA AF scheme can outperform the traditional BF AF scheme, especially when there are many users in the system or many generalized power constraints in the problem formulation. From a practical perspective, the BFA AF scheme offers two degrees of freedom (DoFs) in beamformer design, as opposed to the one DoF offered by the BF AF scheme, to improve the receivers' SINR. In the latter part of this paper, we demonstrate how this extra DoF leads to provable performance gains by considering two special cases of multicasting, where the AF process is shown to employ a special structure. The numerical simulations further validate that the proposed BFA AF scheme outperforms the BF AF scheme and works well for large-scale relay systems.