  Grant-free access is considered as a key enabler to massive machine-type
communications (mMTC) as it promotes energy-efficiency and small signalling
overhead. Due to the sporadic user activity in mMTC, joint user identification
and channel estimation (JUICE) is a main challenge. This paper addresses the
JUICE in single-cell mMTC with single-antenna users and a multi-antenna base
station (BS) under spatially correlated fading channels. In particular, by
leveraging the sporadic user activity, we solve the JUICE in a multi
measurement vector compressed sensing (CS) framework under two different cases,
with and without the knowledge of prior channel distribution information (CDI)
at the BS. First, for the case without prior information, we formulate the
JUICE as an iterative reweighted $\ell_{2,1}$-norm minimization problem.
Second, when the CDI is known to the BS, we exploit the available information
and formulate the JUICE from a Bayesian estimation perspective as a maximum
\emph{a posteriori} probability (MAP) estimation problem. For both JUICE
formulations, we derive efficient iterative solutions based on the alternating
direction method of multipliers (ADMM). The numerical experiments show that the
proposed solutions achieve higher channel estimation quality and activity
detection accuracy with shorter pilot sequences compared to existing
algorithms.
