  Symbol-level precoding (SLP) has recently emerged as a new paradigm for
physical-layer transmit precoding in multiuser multi-input-multi-output (MIMO)
channels. It exploits the underlying symbol constellation structure, which the
conventional paradigm of linear precoding does not, to enhance symbol-level
performance such as symbol error probability (SEP). It allows the precoder to
take a more general form than linear precoding. This paper aims to better
understand the relationships between SLP and linear precoding, subsequent
design implications, and further connections beyond the existing SLP scope. Our
study is built on a basic signal observation, namely, that SLP can be
equivalently represented by a zero-forcing (ZF) linear precoding scheme
augmented with some appropriately chosen symbol-dependent perturbation terms,
and that some extended form of SLP is equivalent to a vector perturbation (VP)
nonlinear precoding scheme augmented with the above-noted perturbation terms.
We examine how insights arising from this perturbed ZF and VP interpretations
can be leveraged to i) substantially simplify the optimization of certain SLP
design criteria, namely, total or peak power minimization subject to SEP
quality guarantees and under quadrature amplitude modulation (QAM)
constellations; and ii) derive heuristic but computationally cheaper SLP
designs. We also touch on the analysis side by showing that, under the total
power minimization criterion, the basic ZF scheme is a near-optimal SLP scheme
when the QAM order is very high--which gives a vital implication that SLP is
more useful for lower-order QAM cases. Numerical results further indicate the
merits and limitations of the different SLP designs derived from the perturbed
ZF and VP interpretations.
