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

Interference Exploitation Precoding for Multi-Level Modulations: Closed-Form Solutions