Abstract: Considering the cyclostationarity of noise radiated by rotating machinery, a microphone array-based non-synchronous measurement method for identifying cyclostationary sound sources is proposed. First, the low-rank property of the cyclic cross-spectral matrix is demonstrated, and the problem of non-synchronous measurement of cyclostationary sound sources is formulated as a low-rank matrix completion problem. Then, leveraging the forward propagation model of cyclostationary sources, a new spatial continuity constraint is defined specifically for cyclostationary sound sources, and a constrained optimization problem is constructed. This problem is solved using the fast iterative shrinkage-thresholding algorithm (FISTA), achieving non-synchronous measurement imaging and identification of cyclostationary sources. The proposed cyclostationary non-synchronous measurement method is validated through numerical simulations and anechoic chamber experiments. The results show that compared to the traditional cyclostationary conventional beamforming method, the proposed approach significantly improves spatial resolution in low-frequency imaging and reduces sidelobe levels in high-frequency imaging, enabling effective identification of cyclostationary sound sources.