  Compressed sensing multi-user detection (CS-MUD) algorithms play a key role
in optimizing grant-free (GF) non-orthogonal multiple access (NOMA) for massive
machine-type communications (mMTC). However, current CS-MUD algorithms cannot
be efficiently parallelized, leading to computationally expensive
implementations of joint activity detection and channel estimation (JADCE) as
the number of deployed machine-type devices (MTDs) increases. To address this,
the present work proposes novel JADCE algorithms that can be applied in
parallel for different clusters of MTDs by exploiting the structure of the
pilot sequences. These are the approximation error method (AEM)-alternating
direction method of multipliers (ADMM), and AEM-sparse Bayesian learning (SBL).
Results presented in terms of the normalized mean square error and the
probability of miss detection show comparable performance to the conventional
algorithms. However, both AEM-ADMM and AEM-SBL algorithms have significantly
reduced computational complexity and run times, thus, facilitating network
scalability.
