Symbol-Level Precoding Through the Lens of Zero Forcing and Vector Perturbation

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.