  Channel estimation for the downlink of frequency division duplex (FDD)
massive MIMO systems is well known to generate a large overhead as the amount
of training generally scales with the number of transmit antennas in a MIMO
system. In this paper, we consider the solution of extrapolating the channel
frequency response from uplink pilot estimates to the downlink frequency band,
which completely removes the training overhead. We first show that conventional
estimators fail to achieve reasonable accuracy. We propose instead to use
high-resolution channel estimation. We derive theoretical lower bounds (LB) for
the mean squared error (MSE) of the extrapolated channel. Assuming that the
paths are well separated, the LB is simplified in an expression that gives
considerable physical insight. It is then shown that the MSE is inversely
proportional to the number of receive antennas while the extrapolation
performance penalty scales with the square of the ratio of the frequency offset
and the training bandwidth. The channel extrapolation performance is validated
through numeric simulations and experimental measurements taken in an anechoic
chamber. Our main conclusion is that channel extrapolation is a viable solution
for FDD massive MIMO systems if accurate system calibration is performed and
favorable propagation conditions are present.
