  In this paper, we investigate the performance of cell-free massive MIMO
systems with massive connectivity. With the generalized approximate message
passing (GAMP) algorithm, we obtain the minimum mean-squared error (MMSE)
estimate of the effective channel coefficients from all users to all access
points (APs) in order to perform joint user activity detection and channel
estimation. Subsequently, using the decoupling properties of MMSE estimation
for large linear systems and state evolution equations of the GAMP algorithm,
we obtain the variances of both the estimated channel coefficients and the
corresponding channel estimation error. Finally, we study the achievable uplink
rates with zero-forcing (ZF) detector at the central processing unit (CPU) of
the cell-free massive MIMO system. With numerical results, we analyze the
impact of the number of pilots used for joint activity detection and channel
estimation, the number of APs, and signal-to-noise ratio (SNR) on the
achievable rates.
