In this work, we consider the design of hybrid analog-digital (HAD) multi-carrier MIMO-OFDM two-way relaying systems, where the relay station is equipped with a HAD amplify-and-forward architecture and every mobile station is equipped with a fully-digital beamforming architecture. We propose a sub-optimal solution by reformulating the original non-convex problem as a constrained Tucker2 decomposition with the objective of minimizing the sum Euclidean-norm between the HAD amplification matrices and their fully-digital counterparts. For the fully-digital amplification matrix design, we use a Frobenius-norm maximization of the effective channels on every subcarrier and propose an effective solution applicable for multi-stream communication scenarios. After that, we propose an alternating maximization (AltMax) HAD solution by exploiting the tensor structure of the reformulated problem. Simulation results are provided, where we show that the proposed fully-digital and AltMax-based HAD amplification matrix designs outperform some benchmark methods, especially for multi-stream communication scenarios.
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