Two-Way Filter-and-Forward Beamforming for Frequency-Selective Channels

In this paper, we consider filter-and-forward beamforming (FF-BF) for two-way relay networks employing single-carrier transmission over frequency-selective channels. In FF-BF, the relay nodes filter the received signal using finite impulse response (FIR) or infinite impulse response (IIR) filters. For the processing at the transceivers, we investigate two different cases: (1) simple slicing without equalization and (2) linear equalization (LE) or decision-feedback equalization (DFE). For the first case, we optimize FIR FF-BF filters, respectively, for maximization of the minimum transceiver signal-to-interference-plus-noise ratio (SINR) subject to a relay transmit power constraint and for minimization of the total relay transmit power subject to two quality of service (QoS) constraints. We show that both problems can be transformed into a convex second-order cone programming (SOCP) problem, which can be efficiently solved using standard tools. For the second case, we optimize IIR and FIR FF-BF filters for max-min optimization of the SINR, and for transceivers with zero-forcing LE, also for minimization of the sum mean-squared error (MSE) at the equalizer outputs of both transceivers. Leveraging results from FF-BF for one-way relaying, we establish an upper and an achievable lower bound for the max-min problem and an exact solution for the sum MSE problem. Since the gap between the upper and the lower bound for the max-min problem is small, a close-to-optimal solution is obtained. Our simulation results reveal that the performance of FF-BF without equalization at the transceivers crucially depends on the slicer decision delay and transceivers with slicers can closely approach the performance of transceivers with equalizers provided that the FF-BF filters are sufficiently long and a sufficient number of relays is deployed.