  In this paper, we study how to efficiently and reliably detect active devices
and estimate their channels in a multiple-input multiple-output (MIMO)
orthogonal frequency-division multiplexing (OFDM) based grant-free
non-orthogonal multiple access (NOMA) system to enable massive machine-type
communications (mMTC). First, by exploiting the correlation of the channel
frequency responses in narrow-band mMTC, we propose a block-wise linear channel
model. Specifically, the continuous OFDM subcarriers in the narrow-band are
divided into several sub-blocks and a linear function with only two variables
(mean and slope) is used to approximate the frequency-selective channel in each
sub-block. This significantly reduces the number of variables to be determined
in channel estimation and the sub-block number can be adjusted to reliably
compensate the channel frequency-selectivity. Second, we formulate the joint
active device detection and channel estimation in the block-wise linear system
as a Bayesian inference problem. By exploiting the block-sparsity of the
channel matrix, we develop an efficient turbo message passing (Turbo-MP)
algorithm to resolve the Bayesian inference problem with near-linear
complexity. We further incorporate machine learning approaches into Turbo-MP to
learn unknown prior parameters. Numerical results demonstrate the superior
performance of the proposed algorithm over state-of-the-art algorithms.
