Disentangling modes with crossover instantaneous frequencies by synchrosqueezed chirplet transforms, from theory to application

Analysis of signals with oscillatory modes with crossover instantaneous frequencies is a challenging problem in time series analysis. One way to handle this problem is lifting the 2-dimensional time-frequency representation to a 3-dimensional representation, called time-frequency-chirp rate (TFC) representation, by adding one extra chirp rate parameter so that crossover frequencies are disentangled in higher dimension. The chirplet transform is an algorithm for this lifting idea, which leads to a TFC representation. However, in practice, we found that it has a strong ``blurring'' effect in the chirp rate axis, which limits its application in real-world data. Moreover, to our knowledge, we have limited mathematical understanding of the chirplet transform in the literature. Motivated by the need for the real-world data analysis, in this paper, we propose the synchrosqueezed chirplet transform (SCT) that enhances the TFC representation given by the chirplet transform. The resulting concentrated TFC representation has high contrast so that one can better distinguish different modes with crossover instantaneous frequencies. The basic idea is to use the phase information in the chirplet transform to determine a reassignment rule that sharpens the TFC representation determined by the chirplet transform. We also analyze the chirplet transform and provide theoretical guarantees of SCT.