  We propose a novel method for massive Multiple-Input Multiple-Output (massive
MIMO) in Frequency Division Duplexing (FDD) systems. Due to the large frequency
separation between Uplink (UL) and Downlink (DL), in FDD systems channel
reciprocity does not hold. Hence, in order to provide DL channel state
information to the Base Station (BS), closed-loop DL channel probing and
Channel State Information (CSI) feedback is needed. In massive MIMO this incurs
typically a large training overhead. For example, in a typical configuration
with M = 200 BS antennas and fading coherence block of T = 200 symbols, the
resulting rate penalty factor due to the DL training overhead, given by max{0,
1 - M/T}, is close to 0. To reduce this overhead, we build upon the well-known
fact that the Angular Scattering Function (ASF) of the user channels is
invariant over frequency intervals whose size is small with respect to the
carrier frequency (as in current FDD cellular standards). This allows to
estimate the users' DL channel covariance matrix from UL pilots without
additional overhead. Based on this covariance information, we propose a novel
sparsifying precoder in order to maximize the rank of the effective sparsified
channel matrix subject to the condition that each effective user channel has
sparsity not larger than some desired DL pilot dimension T_{dl}, resulting in
the DL training overhead factor max{0, 1 - T_{dl} / T} and CSI feedback cost of
T_{dl} pilot measurements. The optimization of the sparsifying precoder is
formulated as a Mixed Integer Linear Program, that can be efficiently solved.
Extensive simulation results demonstrate the superiority of the proposed
approach with respect to concurrent state-of-the-art schemes based on
compressed sensing or UL/DL dictionary learning.
