  In this letter, we propose a joint active device detection and channel
estimation framework based on factor graphs for asynchronous uplink grant-free
massive multiple-antenna systems. We then develop the message-scheduling GAMP
(MSGAMP) algorithm to perform joint active device detection and channel
estimation. In MSGAMP we apply scheduling techniques based on the residual
belief propagation (RBP) and the activity user detection (AUD) in which
messages are generated using the latest available information. MSGAMP-type
schemes show a good performance in terms of activity error rate and normalized
mean squared error, requiring a smaller number of iterations for convergence
and lower complexity than state-of-the-art techniques.
