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In this paper, we propose closed-form precoding schemes with optimal performance for constructive interference (CI) exploitation in the multiuser multiple-input single-output downlink, where the cases of both strict and non-strict phase rotation are considered. For optimization with strict phase rotation, we mathematically derive the optimal precoding structure with Lagrangian and Karush-Kuhn-Tucker conditions. By formulating its dual problem, the optimization problem is further shown to be equivalent to a quadratic programming over a simplex, which can be solved more efficiently. We then extend our analysis to the case of non-strict phase rotation, where it is mathematically shown that a K -dimensional optimization for non-strict phase rotation is equivalent to a 2K -dimensional optimization for strict phase rotation in terms of the problem formulation. The connection with the conventional zero-forcing precoding is also discussed. Based on the above-mentioned analysis, we further propose an iterative closed-form scheme to obtain the optimal precoding matrix, where within each iteration a closed-form solution can be obtained. Numerical results validate our analysis and the optimality of the proposed iterative closed-form algorithm, and further show that the proposed iterative closed-form scheme offers a flexible performance-complexity tradeoff by limiting the maximum number of iterations, which motivates the use of CI precoding in practical wireless systems.

Interference Exploitation Precoding Made Practical: Optimal Closed-Form Solutions for PSK Modulations