  Currently, orthogonal time frequency space (OTFS) modulation has drawn much
attention to reliable communications in high-mobility scenarios. This paper
proposes a doubly-iterative sparsified minimum mean square error (DI-S-MMSE)
turbo equalizer, which iteratively exchanges the extrinsic information between
a soft-input-soft-input (SISO) MMSE estimator and a SISO decoder. Our proposed
equalizer does not suffer from short loops and approaches the performance of
the near-optimal symbol-wise maximum a posteriori (MAP) algorithm. To exploit
the inherent sparsity of OTFS system, we resort to graph theory to investigate
the sparsity pattern of the channel matrix, and propose two sparsification
guidelines to reduce the complexity of calculating the matrix inverse at the
MMSE estimator. Then, we apply two iterative algorithms to MMSE estimation,
i.e., the Generalized Minimal Residual (GMRES) and Factorized Sparse
Approximate Inverse (FSPAI) algorithms. The former is used at the initial turbo
iteration, whose global convergence is proven in our equalizer, while the
latter is used at the subsequent turbo iterations with the help of our proposed
guidelines. Simulation results demonstrate that our equalizer has a linear
order of complexity while the performance loss incurred by the sparsification
is only 0.2 dB at $10^{-4}$ bit error rate.
