[Paper Review] Disentangling modes with crossover instantaneous frequencies by synchrosqueezed chirplet transforms, from theory to application

Ziyu Chen, Hau‐Tieng Wu|arXiv (Cornell University)|Dec 3, 2021

Image and Signal Denoising Methods4 citations

TL;DR

This paper proposes the synchrosqueezed chirplet transform (SCT), a novel time-frequency-chirp rate (TFC) representation that disentangles oscillatory modes with crossover instantaneous frequencies by leveraging phase reassignment. By enhancing the chirplet transform's blurring effect through phase-based reassignment, SCT achieves high-contrast, concentrated TFC representations, enabling accurate separation of overlapping modes in nonstationary signals with theoretical guarantees.

ABSTRACT

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.

Motivation & Objective

• To address the challenge of disentangling oscillatory modes with crossing instantaneous frequencies in time series analysis.
• To overcome the strong blurring effect in the chirp rate axis of the conventional chirplet transform in real-world data.
• To provide a mathematically rigorous framework for the chirplet transform and its synchrosqueezed variant.
• To develop a high-contrast, concentrated TFC representation that enables accurate mode separation despite frequency crossings.
• To extend the applicability of time-frequency analysis to nonstationary signals with complex, overlapping oscillatory components.

Proposed method

• Lift the 2D time-frequency representation to a 3D time-frequency-chirp rate (TFC) representation using the chirplet transform, which adds a chirp rate parameter to resolve frequency crossovers.
• Apply phase-based reassignment to the chirplet transform output, using the instantaneous phase to determine a new, concentrated TFC representation.
• Define the synchrosqueezed chirplet transform (SCT) as a reassignment of the chirplet transform's time-frequency-chirp rate components based on phase derivative information.
• Use the $ε$-ICT (epsilon-instantaneous chirp rate) model to theoretically justify the method under local quadratic phase assumptions.
• Implement the SCT algorithm with adaptive windowing and validate its performance on synthetic and real-world signals, including wolf howling recordings.
• Establish theoretical convergence and stability guarantees for SCT under the $ε$-ICT assumption, extending existing SST theory to higher-order time-frequency representations.

Figure 1. Top row, from left to right: the plot of $\Re(f_{1}+f_{2})$ , the plot of the IFs of $f_{1}$ and $f_{2}$ , and the spectrogram. Note that their IFs have a crossing point at $(t_{0},\xi_{0})=(3,24)$ . All the plots are generated with the kernel $g_{0}=e^{-\pi x^{2}}$ . Second row, from left

Experimental results

Research questions

• RQ1 Can the chirplet transform be enhanced to reduce blurring in the chirp rate dimension for better mode separation?
• RQ2 How can phase information from the chirplet transform be used to reassign energy in the TFC representation to achieve sharper concentration?
• RQ3 What theoretical conditions ensure the stability and accuracy of the synchrosqueezed chirplet transform in separating modes with crossover instantaneous frequencies?
• RQ4 How does the proposed SCT perform on real-world signals with complex, non-sinusoidal, and overlapping oscillatory components?
• RQ5 What are the computational and theoretical limitations of SCT when applied to signals with non-polynomial or non-quadratic phase functions?

Key findings

• The synchrosqueezed chirplet transform (SCT) successfully disentangles modes with crossover instantaneous frequencies in synthetic signals, as demonstrated by clear separation in the TFC representation.
• SCT achieves significantly higher contrast in the TFC representation compared to the original chirplet transform, which suffers from strong blurring in the chirp rate axis.
• The method effectively resolves frequency crossovers in a synthetic signal composed of two quadratic-phase components, with a crossing point at $ t_0 = 3 $ and $ \xi_0 = 24 $.
• On real-world data, such as a wolf howling signal between 16–17 seconds, SCT clearly separates multiple modes, with visual evidence from TFC plots showing concentrated energy ridges.
• Theoretical analysis confirms that SCT provides stable and concentrated TFC representations under the $\epsilon$-ICT assumption, extending the framework of synchrosqueezing to higher-order time-frequency representations.
• While noise robustness is empirically observed (e.g., in Figure 8), the paper notes that theoretical justification for noise resilience remains an open problem for future work.

Figure 2. An illustration of SCT with the standard Gaussian window. From left to right: the 3-dim visualization of $\mathinner{\!\left\lvert S_{f}^{(g_{0})}(t,\xi,\lambda)\right\rvert}$ , the projection of $\mathinner{\!\left\lvert S_{f}^{(g_{0})}(t,\xi,\lambda)\right\rvert}$ onto the time-frequency