Cooperative Filter-and-Forward Beamforming for Frequency-Selective Channels with Equalization

Most of the existing literature on cooperative relay networks has focused on frequency-nonselective channels or frequency-selective channels with multi-carrier transmission. However, several practical systems employ single-carrier transmission over frequency-selective channels and the design of corresponding relaying schemes is a largely under-explored topic. 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, we assume that the destination node is equipped with a simple linear or decision feedback equalizer. The FF-BF filters at the relays are optimized for maximization of the signal-to-noise ratio at the equalizer output under a joint relay power constraint. For infinite impulse response (IIR) FF-BF filters, we derive a unified expression for the filter frequency response valid for linear equalization, decision feedback equalization, and an idealized matched filter receiver. A numerical algorithm with guaranteed convergence is developed for optimization of the power allocation factor included in the expression for the IIR FF-BF filter frequency response. We also provide an efficient gradient algorithm for recursive calculation of near-optimal finite impulse response (FIR) FF-BF filters. Simulation results show that, in general, short FIR FF-BF filters are sufficient to closely approach the performance of IIR FF-BF filters even in severely frequency-selective channels and that the proposed FF-BF scheme with equalization at the destination achieves substantial performance gains compared to a previously proposed FF-BF scheme without equalization.