  We consider the problem of uplink channel estimation for millimeter wave
(mmWave) systems, where the base station (BS) and mobile stations (MSs) are
equipped with large antenna arrays to provide sufficient beamforming gain for
outdoor wireless communications. Hybrid analog and digital beamforming
structures are employed by both the BS and the MS due to hardware constraints.
We propose a layered pilot transmission scheme and a CANDECOMP/PARAFAC (CP)
decomposition-based method for joint estimation of the channels from multiple
users (i.e. MSs) to the BS. The proposed method exploits the sparse scattering
nature of the mmWave channel and the intrinsic multi-dimensional structure of
the multiway data collected from multiple modes. The uniqueness of the CP
decomposition is studied and sufficient conditions for essential uniqueness are
obtained. The conditions shed light on the design of the beamforming matrix,
the combining matrix and the pilot sequences, and meanwhile provide general
guidelines for choosing system parameters. Our analysis reveals that our
proposed method can achieve a substantial training overhead reduction by
employing the layered pilot transmission scheme. Simulation results show that
the proposed method presents a clear advantage over a compressed sensing-based
method in terms of both estimation accuracy and computational complexity.
