  We consider the problem of channel estimation for millimeter wave (mmWave)
systems, where, to minimize the hardware complexity and power consumption, an
analog transmit beamforming and receive combining structure with only one radio
frequency (RF) chain at the base station (BS) and mobile station (MS) is
employed. Most existing works for mmWave channel estimation exploit sparse
scattering characteristics of the channel. In addition to sparsity, mmWave
channels may exhibit angular spreads over the angle of arrival (AoA), angle of
departure (AoD), and elevation domains. In this paper, we show that angular
spreads give rise to a useful low-rank structure that, along with the sparsity,
can be simultaneously utilized to reduce the sample complexity, i.e. the number
of samples needed to successfully recover the mmWave channel. Specifically, to
effectively leverage the joint sparse and low-rank structure, we develop a
two-stage compressed sensing method for mmWave channel estimation, where the
sparse and low-rank properties are respectively utilized in two consecutive
stages, namely, a matrix completion stage and a sparse recovery stage. Our
theoretical analysis reveals that the proposed two-stage scheme can achieve a
lower sample complexity than a direct compressed sensing method that exploits
only the sparse structure of the mmWave channel. Simulation results are
provided to corroborate our theoretical results and to show the superiority of
the proposed two-stage method.
