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.
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