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High-Order Synchrosqueezed Chirplet Transforms for Multicomponent Signal Analysis

Yi-Ju Yen, De-Yan Lu, Sing-Yuan Yeh, Jian-Jiun Ding, Chun-Yen Shen

TL;DR

This work addresses the challenge of analyzing AM-FM multi-component signals with crossing instantaneous frequencies, where existing SCT methods can bias when higher-order chirp modulation is strong. It introduces the High-Order Synchrosqueezed Chirplet Transform (HSCT), which augments the SCT with third-order phase information through stable reassignment ingredients $\theta_f^{(g)}$, $\tilde{\mu}_f^{(g)}$, and $\tilde{\omega}_f^{(g)}$, and provides a practical implementation via derivative-replacement operators. Theoretical justification is given under the $\epsilon$-ICBT model, including a main theorem that bound the estimation errors of the first three phase derivatives, and the method is validated on synthetic signals showing improved energy concentration (lower $H_{\alpha}$) and enhanced mode separation, especially under large chirp-rate variation. The HSCT degenerates to SCT for pure chirp signals, preserving backward compatibility, and the results suggest broad applicability to three-dimensional time-frequency-chirp analysis and potential extensions to wavelet-based synchrosqueezing. Overall, HSCT offers a robust framework for sharper, higher-order TF representations in non-stationary, multi-component signals with varying chirp rates.

Abstract

This study focuses on the analysis of signals containing multiple components with crossover instantaneous frequencies (IF). This problem was initially solved with the chirplet transform (CT). Also, it can be sharpened by adding the synchrosqueezing step, which is called the synchrosqueezed chirplet transform (SCT). However, we found that the SCT goes wrong with the high chirp modulation signal due to the wrong estimation of the IF. In this paper, we present the improvement of the post-transformation of the CT. The main goal of this paper is to amend the estimation introduced in the SCT and carry out the high-order synchrosqueezed chirplet transform. The proposed method reduces the wrong estimation when facing a stronger variety of chirp-modulated multi-component signals. The theoretical analysis of the new reassignment ingredient is provided. Numerical experiments on some synthetic signals are presented to verify the effectiveness of the proposed high-order SCT.

High-Order Synchrosqueezed Chirplet Transforms for Multicomponent Signal Analysis

TL;DR

This work addresses the challenge of analyzing AM-FM multi-component signals with crossing instantaneous frequencies, where existing SCT methods can bias when higher-order chirp modulation is strong. It introduces the High-Order Synchrosqueezed Chirplet Transform (HSCT), which augments the SCT with third-order phase information through stable reassignment ingredients , , and , and provides a practical implementation via derivative-replacement operators. Theoretical justification is given under the -ICBT model, including a main theorem that bound the estimation errors of the first three phase derivatives, and the method is validated on synthetic signals showing improved energy concentration (lower ) and enhanced mode separation, especially under large chirp-rate variation. The HSCT degenerates to SCT for pure chirp signals, preserving backward compatibility, and the results suggest broad applicability to three-dimensional time-frequency-chirp analysis and potential extensions to wavelet-based synchrosqueezing. Overall, HSCT offers a robust framework for sharper, higher-order TF representations in non-stationary, multi-component signals with varying chirp rates.

Abstract

This study focuses on the analysis of signals containing multiple components with crossover instantaneous frequencies (IF). This problem was initially solved with the chirplet transform (CT). Also, it can be sharpened by adding the synchrosqueezing step, which is called the synchrosqueezed chirplet transform (SCT). However, we found that the SCT goes wrong with the high chirp modulation signal due to the wrong estimation of the IF. In this paper, we present the improvement of the post-transformation of the CT. The main goal of this paper is to amend the estimation introduced in the SCT and carry out the high-order synchrosqueezed chirplet transform. The proposed method reduces the wrong estimation when facing a stronger variety of chirp-modulated multi-component signals. The theoretical analysis of the new reassignment ingredient is provided. Numerical experiments on some synthetic signals are presented to verify the effectiveness of the proposed high-order SCT.
Paper Structure (16 sections, 13 theorems, 110 equations, 7 figures, 4 tables, 1 algorithm)

This paper contains 16 sections, 13 theorems, 110 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary: Background of the SST and the SCT.
  3. Proposed Method
  4. Computation for Implementation
  5. Proposed Algorithm
  6. Experimental Results
  7. The Results for Large Chirp-Rate Variation
  8. Separating Signals
  9. Degenerated into the SCT.
  10. More than Two Modes.
  11. Conclusion
  12. Metrics for Implement
  13. Rényi Entropy
  14. Earth mover's distance
  15. Figures
  16. ...and 1 more sections

Key Result

Proposition 1

Suppose that $T^{(g)}_f$ is the chirplet transform defined in Definition CT. Define if $f(x)$ is of the form $A(x)e^{2\pi i\phi(x)}$ , $\log A(x)=\sum_{j=0}^3 [\log A]^{(j)}(t)\frac{(x-t)^j}{j!}$ , and $\phi(x)$ is a cubic polynomial, then we have

Figures (7)

  • Figure 1: (Left) The numerical value of the derivatives of the phase function from the first to the third orders. (Middle) The EMD of the results given by the CT, the SCT, and the proposed method. (Right) The scattering plot of $|\phi"'(t)|$--EMD.
  • Figure 2: (Top row) The visualization of the time-frequency (TF) representation for $x_2$. From left to right: the ideal TF representation, the result of the CT, the result of the original SCT, and the result of the proposed HSCT. (Bottom row) The $3$-dimensional visualization of the time-frequency-chirp (TFC) representation of $x_2$. From left to right: the ideal TFC representation, the result of the CT, the result of the original SCT, and the result of the proposed method.
  • Figure 3: Magnitude of the analyzed chirp rate at $t = 3$ and $\xi = 24$ of $x_1$. The red dotted lines indicate the accurate positions for the impulses. (Left) The result of the CT. (Middle) The result of the SCT. (Right) The result of the proposed method.
  • Figure 4: Magnitude of the analyzed chirp rate at $t = 3.176$ and $\xi = 15.67$ of $x_3$. The red dotted lines indicate the accurate positions for the impulses. (Left) The result of the CT. (Middle) The result of the SCT. (Right) The result of the proposed HSCT method.
  • Figure 5: (Top row): Visualizations of the TF representations for $x_3$. From left to right: the ideal TF representation, the results of the CT, the original SCT, and the proposed HSCT, respectively. (Bottom row): Three dimensional visualizations of the TFC representations of $x_3$. From left to right: the ideal TFC representation, the results of the CT, the original SCT, and the proposed HSCT, respectively.
  • ...and 2 more figures

Theorems & Definitions (29)

  • Definition 1: Ideal Time-Frequency (TF) Representation
  • Definition 2: Short time Fourier transform (STFT)
  • Definition 3: The second-order synchrosqueezing transform (VSST)
  • Definition 4: Chirplet transforms (CT) mann1992timemann1995chirplet
  • Definition 5: Synchrosqueezed Chirplet Transform (SCT) chen2023disentangling
  • Definition 6: The $j^{th}$ q-operator
  • Definition 7: Proposed Higher-Order SCT
  • Proposition 1: Reassignment ingredients for proposed high-order SCT
  • Remark 1
  • Definition 8: The $\epsilon$-intrinsic cubic type function ($\epsilon$-ICBT)
  • ...and 19 more