  In massive multiple-input multiple-output (MIMO) systems, the large number of
antennas would bring a great challenge for the acquisition of the accurate
channel state information, especially in the frequency division duplex mode. To
overcome the bottleneck of the limited number of radio links in hybrid
beamforming, we utilize the neural networks (NNs) to capture the inherent
connection between the uplink and downlink channel data sets and extrapolate
the downlink channels from a subset of the uplink channel state information. We
study the antenna subset selection problem in order to achieve the best channel
extrapolation and decrease the data size of NNs. The probabilistic sampling
theory is utilized to approximate the discrete antenna selection as a
continuous and differentiable function, which makes the back propagation of the
deep learning feasible. Then, we design the proper off-line training strategy
to optimize both the antenna selection pattern and the extrapolation NNs.
Finally, numerical results are presented to verify the effectiveness of our
proposed massive MIMO channel extrapolation algorithm.
