  This paper considers the massive connectivity problem in an asynchronous
grant-free random access system, where a huge number of devices sporadically
transmit data to a base station (BS) with imperfect synchronization. The goal
is to design algorithms for joint user activity detection, delay detection, and
channel estimation. By exploiting the sparsity on both user activity and
delays, we formulate a hierarchical sparse signal recovery problem in both the
single-antenna and the multiple-antenna scenarios. While traditional compressed
sensing algorithms can be applied to these problems, they suffer high
computational complexity and often require the perfect statistical information
of channel and devices. This paper solves these problems by designing the
Learned Approximate Message Passing (LAMP) network, which belongs to
model-driven deep learning approaches and ensures efficient performance without
tremendous training data. Particularly, in the multiple-antenna scenario, we
design three different LAMP structures, namely, distributed, centralized and
hybrid ones, to balance the performance and complexity. Simulation results
demonstrate that the proposed LAMP networks can significantly outperform the
conventional AMP method thanks to their ability of parameter learning. It is
also shown that LAMP has robust performance to the maximal delay spread of the
asynchronous users.
