  Orthogonal time frequency and space (OTFS) modulation is a promising
technology that satisfies high Doppler requirements for future mobile systems.
OTFS modulation encodes information symbols and pilot symbols into the
two-dimensional (2D) delay-Doppler (DD) domain. The received symbols suffer
from inter-Doppler interference (IDI) in the fading channels with fractional
Doppler shifts that are sampled at noninteger indices in the DD domain. IDI has
been treated as an unavoidable effect because the fractional Doppler shifts
cannot be obtained directly from the received pilot symbols. In this paper, we
provide a solution to channel estimation for fractional Doppler channels. The
proposed estimation provides new insight into the OTFS input-output relation in
the DD domain as a 2D circular convolution with a small approximation.
According to the input-output relation, we also provide a low-complexity
channel equalization method using the estimated channel information. We
demonstrate the error performance of the proposed channel estimation and
equalization in several channels by simulations. The simulation results show
that in high-mobility environments, the total system utilizing the proposed
methods outperforms orthogonal frequency division multiplexing (OFDM) with
ideal channel estimation and a conventional channel estimation method using a
pseudo sequence.
