  In this paper, we study closed-form interference-exploitation precoding for
multi-level modulations in the downlink of multi-user multiple-input
single-output (MU-MISO) systems. We consider two distinct cases: first, for the
case where the number of served users is not larger than the number of transmit
antennas at the base station (BS), we mathematically derive the optimal
precoding structure based on the Karush-Kuhn-Tucker (KKT) conditions. By
formulating the dual problem, the precoding problem for multi-level modulations
is transformed into a pre-scaling operation using quadratic programming (QP)
optimization. We further consider the case where the number of served users is
larger than the number of transmit antennas at the BS. By employing the pseudo
inverse, we show that the optimal solution of the pre-scaling vector is
equivalent to a linear combination of the right singular vectors corresponding
to zero singular values, and derive the equivalent QP formulation. We also
present the condition under which multiplexing more streams than the number of
transmit antennas is achievable. For both considered scenarios, we propose a
modified iterative algorithm to obtain the optimal precoding matrix, as well as
a sub-optimal closed-form precoder. Numerical results validate our derivations
on the optimal precoding structures for multi-level modulations, and
demonstrate the superiority of interference-exploitation precoding for both
scenarios.
