This paper considers hybrid beamforming (HB) for downlink multiuser massive multiple-input multiple-output (MIMO) systems with frequency selective channels. The proposed HB design employs sets of digitally controlled phase (fixed phase) paired phase shifters (PSs) and switches. For this system, first we determine the required number of radio frequency (RF) chains and 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 ≤ N is the rank of the combined digital precoder matrices of all subcarriers. 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 subcarrier. 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.

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