Cooperative 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. For both cases, we optimize FF-BF filters for maximization of the minimum transceiver signal-to-interference-plus-noise ratio (SINR) subject to a relay transmit power constraint. For case (1), we show that the optimization problem for FIR filters can be transformed into a convex second-order cone programming problem, which can be efficiently solved using standard tools. For case (2), leveraging results from IIR FF-BF for one-way relaying, we establish an upper and an achievable lower bound for the max-min problem. Since the gap between the upper and the lower bound 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 FIR FF-BF filters are sufficiently long.