  Full-dimensional (FD) multi-user massive multiple input multiple output
(m-MIMO) systems employ large two-dimensional (2D) rectangular antenna arrays
to control both the azimuth and elevation angles of signal transmission. We
introduce the sum of two outer products of the azimuth and elevation
beamforming vectors having moderate dimensions as a new class of FD
beamforming. We show that this low-complexity class is capable of outperforming
2D beamforming relying on the single outer product of the azimuth and elevation
beamforming vectors. It is also capable of performing close to its FD
counterpart of massive dimensions in terms of either the users minimum rate or
their geometric mean rate (GM-rate), or sum rate (SR). Furthermore, we also
show that even FD beamforming may be outperformed by our outer product-based
improper Gaussian signaling solution. Explicitly, our design is based on
low-complexity algorithms relying on convex problems of moderate dimensions for
max-min rate optimization or on closed-form expressions for GM-rate and SR
maximization.
