We consider a cell-free hybrid massive multiple-input multiple-output (MIMO) system with $K$ users and $M$ access points (APs), each with $N_a$ antennas and $N_r< N_a$ radio frequency (RF) chains. When $Kll M{N_a}$, efficient uplink channel estimation and data detection with reduced number of pilots can be performed based on low-rank matrix completion. However, such a scheme requires the central processing unit (CPU) to collect received signals from all APs, which may enable the CPU to infer the private information of user locations. We therefore develop and analyze privacy-preserving channel estimation schemes under the framework of differential privacy (DP). As the key ingredient of the channel estimator, two joint differentially private noisy matrix completion algorithms based respectively on Frank-Wolfe iteration and singular value decomposition are presented. We provide an analysis on the tradeoff between the privacy and the channel estimation error. In particular, we show that the estimation error can be mitigated while maintaining the same privacy level by increasing the payload size with fixed pilot size; and the scaling laws of both the privacy-induced and privacy-independent error components in terms of payload size are characterized. Simulation results are provided to further demonstrate the tradeoff between privacy and channel estimation performance.
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