Distributed antennas must be phase-calibrated (phase-synchronized) for certain operations, such as reciprocity-based joint coherent downlink beamforming, to work. We use rigorous signal processing tools to analyze the accuracy of calibration protocols that are based on over-the-air measurements between antennas, with a focus on scalability aspects for large systems. We show that (i) for some who-measures-on-whom topologies, the errors in the calibration process are unbounded when the network grows; and (ii) despite that conclusion, it is optimal -- irrespective of the topology -- to solve a single calibration problem for the entire system and use the result everywhere to support the beamforming. The analyses are exemplified by investigating specific topologies, including lines, rings, and two-dimensional surfaces.

翻译：对于某些操作（例如基于互易性的联合相干下行波束赋形），分布式天线必须进行相位校准（相位同步）。我们利用严谨的信号处理工具分析基于天线间空间测量校准协议的精度，重点关注大规模系统的可扩展性。我们证明：（i）对于某些“谁测量谁”拓扑结构，当网络规模增大时校准过程误差无界；（ii）尽管存在这一结论，但无论拓扑结构如何，最优策略是为整个系统求解单一校准问题，并将结果全域应用于支撑波束赋形。我们通过分析线型、环型及二维曲面等具体拓扑结构对上述结论进行实例验证。