Per-Relay Power Minimization for Multi-user Multi-channel Cooperative Relay Beamforming

We investigate the optimal relay beamforming problem for multi-user peer-to-peer communication with amplify-and-forward relaying in a multi-channel system. Assuming each source-destination (S-D) pair is assigned an orthogonal channel, we formulate the problem as a min-max per-relay power minimization problem with minimum signal-to-noise (SNR) guarantees. After showing that strong Lagrange duality holds for this nonconvex problem, we transform its Lagrange dual problem to a semi-definite programming problem and obtain the optimal relay beamforming vectors. We identify that the optimal solution can be obtained in three cases, depending on the values of the optimal dual variables. These cases correspond to whether the minimum SNR requirement at each S-D pair is met with equality, and whether the power consumption at a relay is the maximum among relays at optimality. We obtain a semi-closed form solution structure of relay beam vectors, and propose an iterative approach to determine relay beam vector for each S-D pair. We further show that the reverse problem of maximizing the minimum SNR with per-relay power budgets can be solved using our proposed algorithm with an iterative bisection search. Through simulation, we analyze the effect of various system parameters on the performance of the optimal solution. Furthermore, we investigated the effect of imperfect channel side information of the second hop on the performance and quantify the performance loss due to either channel estimation error or limited feedback.