  Configuring the hybrid precoders and combiners in a millimeter wave (mmWave)
multiuser (MU) multiple-input multiple-output (MIMO) system is challenging in
frequency selective channels. In this paper, we develop a system that uses
compressive estimation on the uplink to configure precoders and combiners for
the downlink (DL). In the first step, the base station (BS) simultaneously
estimates the channels from all the mobile stations (MSs) on each subcarrier.
To reduce the number of measurements required, compressed sensing techniques
are developed that exploit common support on the different subcarriers. In the
second step, exploiting reciprocity and the channel estimates, the base station
designs hybrid precoders and combiners. Two algorithms are developed for this
purpose, with different performance and complexity tradeoffs: 1) a
factorization of the purely digital solution, and 2) an iterative hybrid
design. Extensive numerical experiments evaluate the proposed solutions
comparing to state-of-the-art strategies, and illustrating design tradeoffs in
overhead, complexity, and performance.
