  This paper presents a mutual coupling based calibration method for
time-division-duplex massive MIMO systems, which enables downlink precoding
based on uplink channel estimates. The entire calibration procedure is carried
out solely at the base station (BS) side by sounding all BS antenna pairs. An
Expectation-Maximization (EM) algorithm is derived, which processes the
measured channels in order to estimate calibration coefficients. The EM
algorithm outperforms current state-of-the-art narrow-band calibration schemes
in a mean squared error (MSE) and sum-rate capacity sense. Like its
predecessors, the EM algorithm is general in the sense that it is not only
suitable to calibrate a co-located massive MIMO BS, but also very suitable for
calibrating multiple BSs in distributed MIMO systems.
  The proposed method is validated with experimental evidence obtained from a
massive MIMO testbed. In addition, we address the estimated narrow-band
calibration coefficients as a stochastic process across frequency, and study
the subspace of this process based on measurement data. With the insights of
this study, we propose an estimator which exploits the structure of the process
in order to reduce the calibration error across frequency. A model for the
calibration error is also proposed based on the asymptotic properties of the
estimator, and is validated with measurement results.
