High-order synchrosqueezed wavelet-chirplet transform for instantaneous frequency and chirprate estimation

The separation of multicomponent signals with crossing instantaneous frequency (IF) curves remains a fundamental challenge in time-frequency analysis. Although the synchrosqueezed wavelet-chirplet transform (SWCT) enhances time-frequency readability by introducing a chirprate variable, its effectiveness is constrained by the underlying assumption of local linear chirp. Consequently, this method does not perform well when analyzing signals characterized by strong frequency modulation. This paper extends the SWCT framework by relaxing the linear chirp assumption. We model signal components as having polynomial phase behavior over short intervals and derive compact expressions for high-order IF and chirprate reassignment operators. The proposed high-order synchrosqueezed wavelet-chirplet transform (HSWCT) enables accurate estimation of both IF and chirprate, and supports robust mode retrieval even with intersecting IF curves. Another key contribution is a rigorous mathematical analysis of the approximation errors of arbitrary-order reassignment operators for IF and chirprate estimation. When the chirprate vanishes, HSWCT simplifies to the traditional high-order synchrosqueezed wavelet transform. To our best knowledge, no theoretical analysis exists in the literature on the approximation of arbitrary-order SST IF reassignment operators to the IF. As a by-product of this work, our established theorem provides such an analysis, thereby filling a gap in the theoretical framework of high-order SSTs.