Hierarchical Group Method for Over-the-Air Channel Calibration in Cell-Free Massive MIMO

Abstract

Cell-Free massive multiple-input multiple-output (CF-mMIMO) is a key technology for 6G networks, enabling efficient downlink beamforming by exploiting the channel reciprocity under time-division duplex (TDD) operation. However, mismatched radio-frequency (RF) gains in transceiver chains render the composite channel non-reciprocal. In large-scale distributed deployments, calibration faces two fundamental challenges: the accumulation of inter-antenna errors inherent to serial schemes and the prohibitive computational complexity of global optimization. To overcome this trade-off, we propose a novel hierarchical group calibration (HGC) framework. By employing a scalable two-hop architecture, HGC physically interrupts cumulative error chains while avoiding the prohibitive complexity of global methods. We derive a closed-form mean squared error (MSE) expression that explicitly captures the joint impact of geometric distances and hardware imperfections. Guided by this analysis, we develop a topology-aware hierarchical grouping algorithm (HGA) to iteratively optimize reference selection and partitioning. Numerical results confirm that HGC achieves superior performance and robustness compared to conventional benchmarks.