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In time-division duplexing (TDD) systems, massive multiple-inputmultiple-output (MIMO) relies on the channel reciprocity to obtain the downlink(DL) channel state information (CSI) with the uplink (UL) CSI. In practice, themismatches in the radio frequency (RF) analog circuits among different antennasat the base station (BS) break the end-to-end UL and DL channel reciprocity.Antenna calibration is necessary to avoid the severe performance degradationwith massive MIMO. Many calibration schemes are available to compensate the RFgain mismatches and restore the channel reciprocity in TDD massive MIMOsystems. In this paper, we focus on the internal self-calibration scheme wheredifferent BS antennas are interconnected via hardware transmission lines.First, we study the resulting calibration performance for an arbitraryinterconnection strategy. Next, we obtain closed-form Cramer-Rao lower bound(CRLB) expressions for each interconnection strategy at the BS with only (M-1)transmission lines and M denotes the total number of BS antennas. Basing on thederived results, we further prove that the star interconnection strategy isoptimal for internal self-calibration due to its lowest CRLB. In addition, wealso put forward efficient recursive algorithms to derive the correspondingmaximum-likelihood (ML) estimates of all the calibration coefficients.Numerical simulation results are also included to corroborate our theoreticalanalyses and results.

Interconnection Strategies for Self-Calibration of Large Scale Antenna Arrays