  This paper considers hybrid beamforming (HB) for downlink multiuser massive
multiple input multiple output (MIMO) systems with frequency selective
channels. For this system, first we determine the required number of radio
frequency (RF) chains and phase shifters (PSs) such that the proposed HB
achieves the same performance as that of the digital beamforming (DB) which
utilizes $N$ (number of transmitter antennas) RF chains. We show that the
performance of the DB can be achieved with our HB just by utilizing $r_t$ RF
chains and $2r_t(N-r_t + 1)$ PSs, where $r_t \leq N$ is the rank of the
combined digital precoder matrices of all sub-carriers. Second, we provide a
simple and novel approach to reduce the number of PSs with only a negligible
performance degradation. Numerical results reveal that only $20-40$ PSs per RF
chain are sufficient for practically relevant parameter settings. Finally, for
the scenario where the deployed number of RF chains $(N_a)$ is less than $r_t$,
we propose a simple user scheduling algorithm to select the best set of users
in each sub-carrier. Simulation results validate theoretical expressions, and
demonstrate the superiority of the proposed HB design over the existing HB
designs in both flat fading and frequency selective channels.
