  Efficient and low-latency wireless connectivity between the base station (BS)
and a sparse set of sporadically active devices from a massive number of
devices is crucial for random access in emerging massive machine-type
communications (mMTC). This paper addresses the challenge of identifying active
devices while meeting stringent access delay and reliability constraints in
mMTC environments. A novel multi-stage active device identification framework
is proposed where we aim to refine a partial estimate of the active device set
using feedback and hypothesis testing across multiple stages eventually leading
to an exact recovery of active devices after the final stage of processing. In
our proposed approach, active devices independently transmit binary preambles
during each stage, leveraging feedback signals from the BS, whereas the BS
employs a non-coherent binary energy detection. The minimum user identification
cost associated with our multi-stage non-coherent active device identification
framework with feedback, in terms of the required number of channel-uses, is
quantified using information-theoretic techniques in the asymptotic regime of
total number of devices $\ell$ when the number of active devices $k$ scales as
$k = {\Theta}(1)$. Practical implementations of our multi-stage active device
identification schemes, leveraging Belief Propagation (BP) techniques, are also
presented and evaluated. Simulation results show that our multi-stage BP
strategies exhibit superior performance over single-stage strategies, even when
considering overhead costs associated with feedback and hypothesis testing.
