  We consider the problem of downlink training and channel estimation in
frequency division duplex (FDD) massive MIMO systems, where the base station
(BS) equipped with a large number of antennas serves a number of single-antenna
users simultaneously. To obtain the channel state information (CSI) at the BS
in FDD systems, the downlink channel has to be estimated by users via downlink
training and then fed back to the BS. For FDD large-scale MIMO systems, the
overhead for downlink training and CSI uplink feedback could be prohibitively
high, which presents a significant challenge. In this paper, we study the
behavior of the minimum mean-squared error (MMSE) estimator when the channel
covariance matrix has a low-rank or an approximate low-rank structure. Our
theoretical analysis reveals that the amount of training overhead can be
substantially reduced by exploiting the low-rank property of the channel
covariance matrix. In particular, we show that the MMSE estimator is able to
achieve exact channel recovery in the asymptotic low-noise regime, provided
that the number of pilot symbols in time is no less than the rank of the
channel covariance matrix. We also present an optimal pilot design for the
single-user case, and an asymptotic optimal pilot design for the multi-user
scenario. Lastly, we develop a simple model-based scheme to estimate the
channel covariance matrix, based on which the MMSE estimator can be employed to
estimate the channel. The proposed scheme does not need any additional training
overhead. Simulation results are provided to verify our theoretical results and
illustrate the effectiveness of the proposed estimated covariance-assisted MMSE
estimator.
