  This paper addresses the joint user identification and channel estimation
(JUICE) problem in machine-type communications under the practical spatially
correlated channels model with unknown covariance matrices. Furthermore, we
consider an MTC network with hierarchical user activity patterns following an
event-triggered traffic mode. Therein the users are distributed over clusters
with a structured sporadic activity behaviour that exhibits both cluster-level
and intra-cluster sparsity patterns. To solve the JUICE problem, we first
leverage the concept of strong priors and propose a
hierarchical-sparsity-inducing spike-and-slab prior to model the structured
sparse activity pattern. Subsequently, we derive a Bayesian inference scheme by
coupling the expectation propagation (EP) algorithm with the expectation
maximization (EM) framework. Second, we reformulate the JUICE as a maximum a
posteriori (MAP) estimation problem and propose a computationally-efficient
solution based on the alternating direction method of multipliers (ADMM). More
precisely, we relax the strong spike-and-slab prior with a
cluster-sparsity-promoting prior based on the long-sum penalty. We then derive
an ADMM algorithm that solves the MAP problem through a sequence of closed-form
updates. Numerical results highlight the significant performance significant
gains obtained by the proposed algorithms, as well as their robustness against
various assumptions on the users sparse activity behaviour.
