Matching synchrosqueezing transform: A useful tool for characterizing signals with fast varying instantaneous frequency and application to machine fault diagnosis

Synchrosqueezing transform (SST) can effectively improve the readability of the time-frequency (TF) representation (TFR) of nonstationary signals composed of multiple components with slow varying instantaneous frequency (IF). However, for signals composed of multiple components with fast varying IF, SST still suffers from TF blurs. In this paper, we introduce a time-frequency analysis (TFA) method called matching synchrosqueezing transform (MSST) that achieves a highly concentrated TF representation comparable to the standard TF reassignment methods (STFRM), even for signals with fast varying IF, and furthermore, MSST retains the reconstruction benefit of SST. MSST captures the philosophy of STFRM to simultaneously consider time and frequency variables, and incorporates three estimators (i.e., the IF estimator, the group delay estimator, and a chirp-rate estimator) into a comprehensive and accurate IF estimator. In this paper, we first introduce the motivation of MSST with three heuristic examples. Then we introduce a precise mathematical definition of a class of chirp-like intrinsic-mode-type functions that locally can be viewed as a sum of a reasonably small number of approximate chirp signals, and we prove that MSST does indeed succeed in estimating chirp-rate and IF of arbitrary functions in this class and succeed in decomposing these functions. Furthermore, we describe an efficient numerical algorithm for the practical implementation of the MSST, and we provide an adaptive IF extraction method for MSST reconstruction. Finally, we verify the effectiveness of the MSST in practical applications for machine fault diagnosis, including gearbox fault diagnosis for a wind turbine in variable speed conditions and rotor rub-impact fault diagnosis for a dual-rotor turbofan engine.