  Time-division-duplexing massive multiple-input multiple-output (MIMO) systems
estimate the channel state information (CSI) by leveraging the uplink-downlink
channel reciprocity, which is no longer valid when the mismatch arises from the
asymmetric uplink and downlink radio frequency (RF) chains. Existing works
treat the reciprocity mismatch as constant for simplicity. However, the
practical RF chain consists of nonlinear components, which leads to nonlinear
reciprocity mismatch. In this work, we examine the impact and the calibration
approach of the nonlinear reciprocity mismatch in massive MIMO systems. To
evaluate the impact of the nonlinear mismatch, we first derive the closed-form
expression of the ergodic achievable rate. Then, we analyze the performance
loss caused by the nonlinear mismatch to show that the impact of the mismatch
at the base station (BS) side is much larger than that at the user equipment
side. Therefore, we propose a calibration method for the BS. During the
calibration, polynomial function is applied to approximate the nonlinear
mismatch factor, and over-the-air training is employed to estimate the
polynomial coefficients. After that, the calibration coefficients are computed
by maximizing the downlink achievable rate. Simulation results are presented to
verify the analytical results and to show the performance of the proposed
calibration approach.
