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

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.