  This paper considers the massive connectivity application in which a large
number of potential devices communicate with a base-station (BS) in a sporadic
fashion. The detection of device activity pattern together with the estimation
of the channel are central problems in such a scenario. Due to the large number
of potential devices in the network, the devices need to be assigned
non-orthogonal signature sequences. The main objective of this paper is to show
that by using random signature sequences and by exploiting sparsity in the user
activity pattern, the joint user detection and channel estimation problem can
be formulated as a compressed sensing single measurement vector (SMV) problem
or multiple measurement vector (MMV) problem, depending on whether the BS has a
single antenna or multiple antennas, and be efficiently solved using an
approximate message passing (AMP) algorithm. This paper proposes an AMP
algorithm design that exploits the statistics of the wireless channel and
provides an analytical characterization of the probabilities of false alarm and
missed detection by using the state evolution. We consider two cases depending
on whether the large-scale component of the channel fading is known at the BS
and design the minimum mean squared error (MMSE) denoiser for AMP according to
the channel statistics. Simulation results demonstrate the substantial
advantage of exploiting the statistical channel information in AMP design;
however, knowing the large-scale fading component does not offer tangible
benefits. For the multiple-antenna case, we employ two different AMP
algorithms, namely the AMP with vector denoiser and the parallel AMP-MMV, and
quantify the benefit of deploying multiple antennas at the BS.
