Stochastic Robust Collaborative Beamforming: Non-Regenerative Relay

In this paper, a novel stochastic robust method is proposed for multiuser distributed relay beamforming systems. All of the channels are subject to complex Gaussian uncertainty. The robust problem is formulated to minimize the total power transmission of the relays, which are subject to the outage probabilistic quality of service (QoS) constraint at each receiver. Based on a new quadratic type of confidence interval inequality for the quadratic form of the Gaussian random variable, the original outage-based problem is reformulated to a nonconvex semidefinite programming (SDP) with a rank constraint. By using the semidefinite relaxation technique, this issue turns into a convex optimization problem, consisting of mixed semidefinite and conic quadratic constraints. A customized stochastic programming technique is proposed to obtain a solution to the original problem. Three other competitive approaches are numerically compared to the proposed method, which are: 1) the robust stochastic method which is based on the central limit theorem (CLT) approximation; 2) the Berenstain inequality approach; and 3) the worst-case robust approach, which is adapted to our stochastic scenario. The first and second methods do not meet the outage constraints whereas the third method qualifies the constraints but it is outperformed by our proposed method in terms of its transmission power.