  In this paper, we study the interference exploitation precoding for the
scenario where the number of streams simultaneously transmitted by the base
station (BS) is larger than that of transmit antennas at the BS, and derive the
optimal precoding structure by employing the pseudo inverse. We show that the
optimal pre-scaling vector is equal to a linear combination of the right
singular vectors that correspond to zero singular values of the coefficient
matrix. By formulating the dual problem, the optimal precoding matrix can be
expressed as a function of the dual variables in a closed form, and an
equivalent quadratic programming (QP) formulation is further derived for
computational complexity reduction. Numerical results validate our analysis and
demonstrate significant performance improvements for interference exploitation
precoding for the considered scenario.
