Fast multiscale order-frequency spectral correlation estimator with sparse prior for rotating machinery signals

Recent research indicates that rotating machinery signals often exhibit cyclostationarity under constant speed and angle-time cyclostationarity under variable speed. Correspondingly, two core statistical descriptors—the spectral correlation (SC) and order-frequency spectral correlation (OF-SC)—are estimated via the averaged cyclic periodogram (ACP)-based or cyclic modulation spectrum (CMS)-based estimators. CMS-based estimators are widely favored due to their reduced computational cost compared to ACP-based estimators; however, their performance remains subject to the time–frequency resolution trade-offs inherent to short-time Fourier transform (STFT)-based implementations. In particular, the detectable range of periodic modulation components is limited by the resolution characteristics of the STFT. To address this limitation from a scale-adaptive perspective, the continuous wavelet transform is incorporated into the CMS-based estimators within the cyclostationary framework, resulting in the wavelet CMS (WCMS) estimator for SC estimation. This estimator is further extended to the angle-time cyclostationary framework, yielding the order-frequency WCMS (OF-WCMS) for estimating OF-SC. Moreover, inspired by the prior knowledge that modulation components in many practical rotating machinery signals exhibit sparsity, a modified sparse fast Fourier transform algorithm is integrated to improve computational efficiency along the cyclic frequency or cyclic order axis. This leads to the sparse WCMS (SWCMS) and order-frequency SWCMS (OF-SWCMS) estimators. The proposed estimators are comprehensively evaluated in terms of resolution characteristics, computational complexity, and observable cyclic frequency/order range. Simulation results using typical cyclostationary and angle-time cyclostationary signal models indicate that the proposed estimators can detect a broader range of periodic modulation components while maintaining low computational cost. Two application cases further demonstrate their practical applicability in prognostics and health management for complex rotating machinery within the existing cyclostationary and angle-time cyclostationary frameworks.