  Wireless communications over fast fading channels are challenging, requiring
either frequent channel tracking or complicated signaling schemes such as
orthogonal time frequency space (OTFS) modulation. In this paper, we propose
low-complexity frequency domain equalizations to combat fast fading, based on
novel discrete delay-time and frequency-Doppler channel models. Exploiting the
circular stripe diagonal nature of the frequency-Doppler channel matrix, we
introduce low-complexity frequency domain minimum mean square error (MMSE)
equalization for OTFS systems with fully resolvable Doppler spreads. We also
demonstrate that the proposed MMSE equalization is applicable to conventional
orthogonal frequency division multiplexing (OFDM) and single carrier frequency
domain equalization (SC-FDE) systems with short signal frames and partially
resolvable Doppler spreads. After generalizing the input-output data symbol
relationship, we analyze the equalization performance via channel matrix
eigenvalue decomposition and derive a closed-form expression for the
theoretical bit-error-rate. Simulation results for OTFS, OFDM, and SC-FDE
modulations verify that the proposed low-complexity frequency domain
equalization methods can effectively exploit the time diversity over fast
fading channels. Even with partially resolvable Doppler spread, the
conventional SC-FDE can achieve performance close to OTFS, especially in fast
fading channels with a dominating line-of-sight path.
