Robust Transceiver Design for SC-FDE Multi-hop Full-Duplex Decode-and-Forward Relaying Systems

In this paper, we consider the robust transceiver design for a multi-hop full-duplex decode-and-forward (DF) relay system employing single-carrier transmission with frequency-domain equalization (SC-FDE). We take into account the effect of imperfect channel state information (CSI) where the CSI errors are modeled as Gaussian random variables with known statistics. The design of the precoding at the transmitter and the equalization at the receiver is formulated as an optimization problem with the objective to minimize two relevant performance metrics, namely, the sum mean-square error (MSE) and the maximum MSE across the different hops, subject to separate transmit power constraints for the nodes. We show that the equalization filters can be optimized individually at the receiving nodes and take the form of robust Wiener filters. However, due to the loopback/backward interference, the transmit signals in the different hops are coupled and the transmit precoding problem leads to a nonconvex power allocation problem in the frequency domain. To find the optimal power allocation, we first employ a sequential geometric programming (sGP) approach, which uses the condensation technique to transform the objective function into a posynomial and then solves a sequence of standard GP problems. The sGP approach requires global channel knowledge at a central node and the involved subproblems admit only numerical solutions. To gain further insight into the structure of the problem, we also consider an alternating optimization (AO) approach for power allocation where convex programming problems and difference of convex programming problems are solved in an alternating manner. The resulting AO algorithm admits closed-form solutions in each iteration step and requires less signaling overhead compared to the centralized sGP scheme. Numerical results for the MSEs and achievable rates of the proposed robust schemes are provided, showing that the proposed sGP and AO algorithms yield significant performance gains compared to conventional half-duplex relay systems and nonrobust full-duplex designs.