  This paper proposes a computationally efficient algorithm to solve the joint
data and activity detection problem for massive random access with massive
multiple-input multiple-output (MIMO). The BS acquires the active devices and
their data by detecting the transmitted preassigned nonorthogonal signature
sequences. This paper employs a covariance based approach that formulates the
detection problem as a maximum likelihood estimation (MLE) problem. To
efficiently solve the problem, this paper designs a novel iterative algorithm
with low complexity in the regime where the device activity pattern is sparse
$\unicode{x2013}$ a key feature that existing algorithmic designs have not
previously exploited for reducing complexity. Specifically, at each iteration,
the proposed algorithm focuses on only a small subset of all potential
sequences, namely the active set, which contains a few most likely active
sequences (i.e., transmitted sequences by all active devices), and performs the
detection for the sequences in the active set. The active set is carefully
selected at each iteration based on the current detection result and the
first-order optimality condition of the MLE problem. Simulation results show
that the proposed active set algorithm enjoys significantly better
computational efficiency (in terms of the CPU time) than the state-of-the-art
algorithms.
