Unified IRS-Aided MIMO Transceiver Designs via Majorization Theory

In this paper, we develop a unified framework for IRS-aided transceiver designs under general power constraints in multiple-input multiple-output (MIMO) systems which implement interference (pre-)subtraction via Tomlinson-Harashima precoding (THP) or Decision Feedback Equalization (DFE) technologies. Armed with majorization theory, two fundamental classes of performance criteria, namely K-increasing Schur-concave and Schur-convex functions of the logarithm of Mean Square Error (MSE) of the data stream, are investigated in depth. Firstly, we propose a simplified counterpart of the optimal transceiver design under general power constraints, with equivalence guaranteed by Pareto optimization theory and Lagrange duality. Moreover, the optimal semi-closed form solution to this simplified transceiver design can be attained using the modified subgradient method. Next, we prove that for any Schur-concave objective, the optimal nonlinear THP (DFE) design is in essence the linear precoding (equalization). For any Schur-convex objective, the optimal transceiver design results in individual data streams with equal MSEs, and thereby reduces to the Gaussian mutual information maximization based design. Based on the above conclusions, we further propose an efficient alternating optimization algorithm to decouple the optimization of the transmit precoder and the IRS reflection coefficients, where the classical successive convex approximation (SCA) technique is applied to fight against non-convex subproblems. From the low computational complexity perspective, a two-stage scheme is also developed inspired by the capability of the IRS in constructing favorable wireless links. Finally, numerical results show the global optimality of the modified subgradient method and the excellent performance of the proposed alternating optimization algorithm and two-stage scheme.