  Grant-free non-orthogonal multiple access has been regarded as a viable
approach to accommodate access for a massive number of machine-type devices
with small data packets. The sporadic activation of the devices creates a
multiuser setup where it is suitable to use compressed sensing in order to
detect the active devices and decode their data. We consider asynchronous
access of machine-type devices that send data packets of different frame sizes,
leading to data length diversity. We address the composite problem of activity
detection, channel estimation, and data recovery by posing it as a structured
sparse recovery, having three-level sparsity caused by sporadic activity,
symbol delay, and data length diversity. We approach the problem through
approximate message passing with a backward propagation algorithm (AMP-BP),
tailored to exploit the sparsity, and in particular the data length diversity.
Moreover, we unfold the proposed AMP-BP into a model-driven network, termed
learned AMP-BP (LAMP-BP), which enhances detection performance. The results
show that the proposed LAMP-BP outperforms existing methods in activity
detection and data recovery accuracy.
