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).