  Cell-free (CF) massive multiple-input multiple-output (MIMO) systems show
great potentials in low-mobility scenarios, due to cell boundary disappearance
and strong macro diversity. However, the great Doppler frequency offset (DFO)
leads to serious inter-carrier interference in orthogonal frequency division
multiplexing (OFDM) technology, which makes it difficult to provide
high-quality transmissions for both high-speed train (HST) operation control
systems and passengers. In this paper, we focus on the performance of CF
massive MIMO-OFDM systems with both fully centralized and local minimum mean
square error (MMSE) combining in HST communications. Considering the local
maximum ratio (MR) combining, the large-scale fading decoding (LSFD)
cooperation and the practical effect of DFO on system performance, exact
closed-form expressions for uplink spectral efficiency (SE) expressions are
derived. We observe that cooperative MMSE combining achieves better SE
performance than uncooperative MR combining. In addition, HST communications
with small cell and cellular massive MIMO-OFDM systems are compared in terms of
SE. Numerical results reveal that the CF massive MIMO-OFDM system achieves a
larger and more uniform SE than the other systems. Finally, the train antenna
centric (TA-centric) CF massive MIMO-OFDM system is designed for practical
implementation in HST communications, and three power control schemes are
adopted to optimize the propagation of TAs for reducing the impact of the DFO.
