In this paper, we consider the robust transceiver design for a two-hop full-duplex decode-and-forward relay system employing single-carrier transmission with frequency-domain equalization (SC-FDE). The design problem for the transmit precoding and receive equalization is formulated as an optimization problem with the objective to minimize the sum mean-squared error (MSE) of the two hops subject to separate node transmit power constraints. 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 interference, the transmissions in the two hops are coupled and the transmit precoding problem boils down to a non-convex power allocation problem in the frequency domain. An alternating optimization approach is proposed to obtain the power allocation where convex programming problems and difference of convex programming problems are solved in an alternating manner. Numerical results are provided to validate the MSE and the achievable rate of the proposed robust schemes, showing that significant performance gains can be achieved compared to conventional half-duplex systems and non-robust full-duplex designs.
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