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We derive the optimal signaling for a multiple input multiple output (MIMO) full-duplex (FD) two-way channel under imperfect transmit front-end chains. We characterize two-way rates of the channel by using a game-theoretical approach, where we focus on the Pareto boundary of the achievable rate region and the Nash equilibrium (NE). For a MIMO FD channel, the Pareto boundary achieves the global optimality. However, deriving the Pareto boundary amounts to solving a family of centralized non-convex problems. By introducing auxiliary variables, we decouple and convert the Pareto boundary into a family of convex problems, which enables us to obtain the Pareto boundary with low computational complexity. In a MISO FD two-way channel, we further present a closed-form expression for the Pareto-optimal signaling. In our numerical examples, we quantify gains in the achievable rates of the Pareto-optimal signaling over the zero-forcing beamforming and NE. For a distributed MIMO FD channel, we establish the existence of NE and present a condition for the uniqueness of NE. We then propose an iterative water-filling algorithm, which is capable of reaching the NE. Through simulations, the threshold of the self-interference level is found below which the FD NE outperforms the half-duplex TDMA.

Signaling Design of Two-Way MIMO Full-Duplex Channel: Optimality Under Imperfect Transmit Front-End Chain