  In this paper, the impacts of imperfect channel covariance matrix on the
spectral efficiency (SE) of cell-free distributed massive multiple-input
multiple-output (MIMO) systems are analyzed. We propose to estimate the channel
covariance matrix by alternately using the assigned pilots and their
phase-shifted pilots in different coherent blocks, which improves the accuracy
of channel estimation with imperfect covariance matrix and reduces pilot
overhead. Under this scheme, the closed-form expressions of SE with maximum
ratio combination (MRC) and zero-forcing (ZF) receivers are derived, which
enables us to select key parameters for better system performance. Simulation
results verify the effectiveness of the proposed channel estimation method and
the accuracy of the derived closed-form expressions. When more coherent blocks
are used to estimate the covariance matrix, we can get better system
performance. Moreover, some insightful conclusions are arrived at from the SE
comparisons between different receiving schemes (ZF and MRC) and different
pilot allocation schemes (orthogonal pilot and pilot reuse).
