  Hybrid analog and digital precoding allows millimeter wave (mmWave) systems
to achieve both array and multiplexing gain. The design of the hybrid precoders
and combiners, though, is usually based on knowledge of the channel. Prior work
on mmWave channel estimation with hybrid architectures focused on narrowband
channels. Since mmWave systems will be wideband with frequency selectivity, it
is vital to develop channel estimation solutions for hybrid architectures based
wideband mmWave systems. In this paper, we develop a sparse formulation and
compressed sensing based solutions for the wideband mmWave channel estimation
problem for hybrid architectures. First, we leverage the sparse structure of
the frequency selective mmWave channels and formulate the channel estimation
problem as a sparse recovery in both time and frequency domains. Then, we
propose explicit channel estimation techniques for purely time or frequency
domains and for combined time/frequency domains. Our solutions are suitable for
both SC-FDE and OFDM systems. Simulation results show that the proposed
solutions achieve good channel estimation quality, while requiring small
training overhead. Leveraging the hybrid architecture at the transceivers gives
further improvement in estimation error performance and achievable rates.
