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One of the biggest challenges in operating massive multiple-inputmultiple-output systems is the acquisition of accurate channel stateinformation at the transmitter. To take up this challenge, time division duplexis more favorable thanks to its channel reciprocity between downlink anduplink. However, while the propagation channel over the air is reciprocal, theradio-frequency front-ends in the transceivers are not. Therefore, calibrationis required to compensate the RF hardware asymmetry.  Although various over-the-air calibration methods exist to address the aboveproblem, this paper offers a unified representation of these algorithms,providing a higher level view on the calibration problem, and introducesinnovations on calibration methods. We present a novel family of calibrationmethods, based on antenna grouping, which improve accuracy and speed up thecalibration process compared to existing methods. We then provide theCram\'er-Rao bound as the performance evaluation benchmark and compare maximumlikelihood and least squares estimators. We also differentiate between coherentand non-coherent accumulation of calibration measurements, and point out thatenabling non-coherent accumulation allows the training to be spread in time,minimizing impact to the data service. Overall, these results have specialvalue in allowing to design reciprocity calibration techniques that are bothaccurate and resource-effective.

A Framework for Over-the-air Reciprocity Calibration for TDD Massive MIMO Systems