  Downlink channel estimation in massive MIMO systems is well known to generate
a large overhead in frequency division duplex (FDD) mode as the amount of
training generally scales with the number of transmit antennas. Using instead
an extrapolation of the channel from the measured uplink estimates to the
downlink frequency band completely removes this overhead. In this paper, we
investigate the theoretical limits of channel extrapolation in frequency. We
highlight the advantage of basing the extrapolation on high-resolution channel
estimation. A lower bound (LB) on the mean squared error (MSE) of the
extrapolated channel is derived. A simplified LB is also proposed, giving
physical intuition on the SNR gain and extrapolation range that can be expected
in practice. The validity of the simplified LB relies on the assumption that
the paths are well separated. The SNR gain then linearly improves with the
number of receive antennas while the extrapolation performance penalty
quadratically scales with the ratio of the frequency and the training
bandwidth. The theoretical LB is numerically evaluated using a 3GPP channel
model and we show that the LB can be reached by practical high-resolution
parameter extraction algorithms. Our results show that there are strong
limitations on the extrapolation range than can be expected in SISO systems
while much more promising results can be obtained in the multiple-antenna
setting as the paths can be more easily separated in the delay-angle domain.
