  We investigate the activity detection and channel estimation issues for
cell-free Internet of Things (IoT) networks with massive random access. In each
time slot, only partial devices are active and communicate with neighboring
access points (APs) using non-orthogonal random pilot sequences. Different from
the centralized processing in cellular networks, the activity detection and
channel estimation in cell-free IoT is more challenging due to the distributed
and user-centric architecture. We propose a two-stage approach to detect the
random activities of devices and estimate their channel states. In the first
stage, the activity of each device is jointly detected by its adjacent APs
based on the vector approximate message passing (Vector AMP) algorithm. In the
second stage, each AP re-estimates the channel using the linear minimum mean
square error (LMMSE) method based on the detected activities to improve the
channel estimation accuracy. We derive closed-form expressions for the activity
detection error probability and the mean-squared channel estimation errors for
a typical device. Finally, we analyze the performance of the entire cell-free
IoT network in terms of coverage probability. Simulation results validate the
derived closed-form expressions and show that the cell-free IoT significantly
outperforms the collocated massive MIMO and small-cell schemes in terms of
coverage probability.
