  Symbol-level precoding (SLP) manipulates the transmitted signals to
accurately exploit the multi-user interference (MUI) in the multi-user
downlink. This enables that all the resultant interference contributes to
correct detection, which is the so-called constructive interference (CI). Its
performance superiority comes at the cost of solving a nonlinear optimization
problem on a symbol-by-symbol basis, for which the resulting complexity becomes
prohibitive in realistic wireless communication systems. In this paper, we
investigate low-complexity SLP algorithms for both phase-shift keying (PSK) and
quadrature amplitude modulation (QAM). Specifically, we first prove that the
max-min SINR balancing (SB) SLP problem for PSK signaling is not separable,
which is contrary to the power minimization (PM) SLP problem, and accordingly,
existing decomposition methods are not applicable. Next, we establish an
explicit duality between the PM-SLP and SB-SLP problems for PSK modulation. The
proposed duality facilitates obtaining the solution to the SB-SLP given the
solution to the PM-SLP without the need for one-dimension search, and vice
versa. We then propose a closed-form power scaling algorithm to solve the
SB-SLP via PM-SLP to take advantage of the separability of the PM-SLP. As for
QAM modulation, we convert the PM-SLP problem into a separable equivalent
optimization problem, and decompose the new problem into several simple
parallel subproblems with closed-form solutions, leveraging the proximal
Jacobian alternating direction method of multipliers (PJ-ADMM). We further
prove that the proposed duality can be generalized to the multi-level
modulation case, based on which a power scaling parallel inverse-free algorithm
is also proposed to solve the SB-SLP for QAM signaling. Numerical results show
that the proposed algorithms offer optimal performance with lower complexity
than the state-of-the-art.
