  Cell-free (CF) massive multiple-input multiple-output (MIMO) is considered as
a promising technology for achieving the ultimate performance limit. However,
due to its distributed architecture and low-cost access points (APs), the
signals received at user equipments (UEs) are most likely asynchronous. In this
paper, we investigate the performance of CF massive MIMO systems with
asynchronous reception, including both effects of delay and oscillator phases.
Taking into account the imperfect channel state information caused by phase
asynchronization and pilot contamination, we obtain novel and closed-form
downlink spectral efficiency (SE) expressions with coherent and non-coherent
data transmission schemes, respectively. Simulation results show that
asynchronous reception destroys the orthogonality of pilots and coherent
transmission of data, and thus results in poor system performance. In addition,
getting a highly accurate delay phase is substantial for CF massive MIMO
systems to achieve coherent transmission gain. Moreover, the oscillator phase
of UEs has a larger effect on SE than that of the APs, because the latter can
be significantly reduced by increasing the number of antennas.
