  This paper studies the impact of imperfect channel covariance information on
the uplink (UL) and downlink (DL) spectral efficiencies (SEs) of a
time-division duplexed (TDD) massive multiple-input multiple-output (MIMO)
system. We derive closed-form expressions for the UL and DL average SEs by
considering linear minimum mean squared (LMMSE)-type and element-wise
LMMSE-type channel estimation that represent LMMSE and element-wise LMMSE with
estimated covariance matrices, respectively. The closed-form expressions of
these average SEs are functions of the number of observations used for
estimating the spatial covariance matrices of individual and contaminated
channels of a target user, and thus enable us to select these key parameters to
achieve the desired SE. We present a theoretical analysis of SE behavior for
different values of these parameters, followed by simulations, which also
demonstrate and validate this behavior. Specifically, we present the SEs
computed using estimated covariance matrices and show the accurate agreement
between the theoretical and simulated SEs as functions of the number of
observations for estimating the covariance matrices of individual and
contaminated channels of a user. We also compare these SEs across channel
estimation techniques using analytical and simulation studies.
