Filter-and-forward beamforming for multiple multi-antenna relays

In this paper, we investigate filter-and-forward beamforming (FF-BF) for relay networks employing single-carrier transmission over frequency-selective channels. In contrast to prior work, which concentrated on multiple single-antenna relay nodes, we consider networks employing multiple multi-antenna relay nodes. For the processing at the destination, we investigate two different cases: (1) A simple slicer without equalization and (2) a linear equalizer or a decision-feedback equalizer. For both cases, we optimize the FF-BF matrix filters at the relays for maximization of the signal-to-interference-plus-noise ratio (SINR) under a transmit power constraint, and for the first case we consider additionally optimization of the FF-BF matrix filters for minimization of the total transmit power under an SINR constraint. For the first case, we obtain closed-form solutions for the optimal finite impulse response (FIR) FF-BF matrix filters, whereas for the second case, we provide the optimal solution for infinite impulse response FF-BF matrix filters, and an efficient gradient algorithm for recursive calculation of near-optimal FIR FF-BF matrix filters. Our simulation results reveal that for a given total number of antennas in the network, a small number of multiple-antenna relays can achieve significant performance gains over a large number single-antenna relays.