  Communications in high-mobility environments have caught a lot of attentions
recently. In this paper, fast time-varying channels for massive multiple-input
multiple-output (MIMO) systems are addressed. We derive the exact channel power
spectrum density (PSD) for the uplink from a high-speed railway (HSR) to a base
station (BS) and propose to further reduce the channel time variation via
beamforming network optimization. A large-scale uniform linear array (ULA) is
equipped at the HSR to separate multiple Doppler shifts in angle domain through
high-resolution transmit beamforming. Each branch comprises a dominant Doppler
shift, which can be compensated to suppress the channel time variation, and we
derive the channel PSD and the Doppler spread to assess the residual channel
time variation. Interestingly, the channel PSD can be exactly expressed as the
product of a pattern function and a beam-distortion function. The former
reflects the impact of array aperture and is the converted radiation pattern of
ULA, while the latter depends on the configuration of beamforming directions.
Inspired by the PSD analysis, we introduce a common configurable amplitudes and
phases (CCAP) parameter to optimize the beamforming network, by partly removing
the constant modulus quantized phase constraints of matched filter (MF)
beamformers. In this way, the residual Doppler shifts can be ulteriorly
suppressed, further reducing the residual channel time variation. The optimal
CCAP parameter minimizing the Doppler spread is derived in a closed form.
Numerical results are provided to corroborate both the channel PSD analysis and
the superiority of beamforming network optimization technique.
