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In this paper, we investigate the symbol-level precoding (SLP) design problem in the downlink of a multiuser multiple-input single-output channel. We consider generic two-dimensional constellations with any shape and size, and confine ourselves to one of the main categories of constructive interference regions (CIR), namely distance preserving CIR (DPCIR). We provide a comprehensive study of DPCIRs and derive several properties for these regions. Using these properties, we first show that any signal in a given DPCIR has a norm greater than or equal to the norm of the corresponding constellation point if and only if the convex hull of the constellation contains the origin. It is followed by proving that the power of the noise-free received signal in a DPCIR is a monotonic strictly increasing function of two parameters relating to the infinite Voronoi edges. Using the convex description of DPCIRs and their characteristics, we formulate two design problems, namely the SLP power minimization with signal-to-interference-plus-noise ratio (SINR) constraints, and the SLP SINR balancing problem under max&#x2013;min fairness criterion. The SLP power minimization based on DPCIRs can straightforwardly be written as a quadratic programming. We derive a simplified reformulation of this problem, which is less computationally complex. The SLP max&#x2013;min SINR, however, is non-convex in its original form, and hence difficult to tackle. We propose alternative optimization approaches, including semidefinite programming formulation and block coordinate descent optimization. We discuss and evaluate the loss due to the proposed alternative methods through extensive simulation results.

Symbol-Level Precoding Design Based on Distance Preserving Constructive Interference Regions