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.
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