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