Cooperative Beamforming for Single-Carrier Frequency-Domain Equalization Systems with Multiple Relays

We consider cooperative beamforming (BF) for {block-based single-carrier frequency-domain equalization (SC-FDE) in a wireless network consisting} of one single-antenna source, one single-antenna destination, and multiple multi-antenna relays. Adopting the minimum mean squared error as optimality criterion, {the optimal frequency-domain linear equalization (LE) and decision-feedback equalization (DFE) receivers are derived and corresponding objective functions for relay BF matrix optimization are specified. For a} sum relay power constraint, we obtain the structure of the optimal relay BF matrices in closed form. While the structure of the optimal relay BF matrices is identical {for LE and DFE as well as for an idealized matched filter receiver}, the solution of the remaining power allocation problem depends on the adopted receiver. The power allocation problem is shown to be convex for all considered receivers and an efficient numerical algorithm for finding the optimal power allocation is provided. Furthermore, to reduce complexity, two suboptimal power allocation schemes assigning identical powers to all relays and/or frequencies are proposed and shown to lead to only a small loss in performance and a remarkable robustness against imperfect channel state information.