  In this paper, we investigate the symbol-level precoding (SLP) design problem
in the downlink of a multiuser multiple-input single-output (MISO) channel. We
consider generic constellations with any arbitrary 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 some 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 noiseless received signal lying on 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
properties, 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-min fairness criterion. The SLP
power minimization based on DPCIRs can straightforwardly be written as a
quadratic program (QP). We provide a simplified reformulation of this problem
which is less computationally complex. The SLP max-min SINR, however, is
non-convex in its original form, and hence difficult to tackle. We propose
several alternative optimization approaches, including semidefinite program
(SDP) formulation and block coordinate descent (BCD) optimization. We discuss
and evaluate the loss due to the proposed alternative methods through extensive
simulation results.
