  Beamspace processing is an emerging technique to reduce baseband complexity
in massive multiuser (MU) multiple-input multiple-output (MIMO) communication
systems operating at millimeter-wave (mmWave) and terahertz frequencies. The
high directionality of wave propagation at such high frequencies ensures that
only a small number of transmission paths exist between user equipments and
basestation (BS). In order to resolve the sparse nature of wave propagation,
beamspace processing traditionally computes a spatial discrete Fourier
transform (DFT) across a uniform linear antenna array at the BS where each DFT
output is associated with a specific beam. In this paper, we study optimality
conditions of the DFT for sparsity-based beamspace processing with idealistic
mmWave channel models and realistic channels. To this end, we propose two
algorithms that learn unitary beamspace transforms using an $\ell^4$-norm-based
sparsity measure, and we investigate their optimality theoretically and via
simulations.
