Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Time-reassigned synchrosqueezing frequency-domain chirplet transform for multicomponent signals with intersecting group delay curves

Shuixin Li, Jiecheng Chen, Qingtang Jiang, Lin Li

TL;DR

The paper tackles the challenge of analyzing multicomponent signals with rapidly varying frequencies and intersecting group-delay ($GD$) curves, where traditional time-frequency methods struggle to separate components. It develops a three-dimensional $TF$-$GDD$ representation based on the frequency-domain chirplet transform (FCT) and introduces the time-reassigned synchrosqueezing frequency-domain chirplet transform (TSFCT) together with a frequency-domain group signal separation operation (FGSSO) for accurate mode retrieval. The authors derive GD and $GDD$ reference functions with explicit error bounds and prove reconstruction error bounds for FGSSO, demonstrating sharp GD/GDD estimation and robust mode recovery in both simulated crossing-GD scenarios and real audio data. This framework yields sharper time-frequency representations and improved component separation for broadband transient signals, with practical implications for audio processing and bioacoustic analysis.

Abstract

To analyze signals with rapid frequency variations or transient components, the time-reassigned synchrosqueezing transform (TSST) and its variants have been recently proposed. Unlike the traditional synchrosqueezing transform, TSST squeezes the time-frequency (TF) coefficients along the group delay (GD) trajectories rather than the instantaneous frequency trajectories. Although TSST methods perform well in analyzing transient signals, they are fundamentally limited in processing multicomponent signals with intersecting GD curves. This limitation compromises the accuracy of both feature extraction and signal component recovery, thereby significantly reducing the interpretability of time-frequency representations (TFRs). This is particularly problematic in broadband signal processing systems, where the linearity of the phase response is critical and precise measurement of group delay dispersion (GDD) is essential. Motivated by the superior capability of frequency-domain signal modeling in characterizing rapidly frequency-varying signals, this paper proposes a novel three-dimensional time-frequency-group delay dispersion (TF-GDD) representation based on the frequency-domain chirplet transform. A subsequent time-reassigned synchrosqueezing frequency-domain chirplet transform (TSFCT) is introduced to achieve a sharper TF-GDD distribution and more accurate GD estimation. For mode retrieval, a novel frequency-domain group signal separation operation (FGSSO) is proposed.The theoretical contributions include a derivation of the approximation error for the GD and GDD reference functions and an establishment of the error bounds for FGSSO-based mode retrieval. Experimental results demonstrate that the proposed TSFCT and FGSSO effectively estimate GDs and retrieve modes--even for modes with intersecting GD trajectories.

Time-reassigned synchrosqueezing frequency-domain chirplet transform for multicomponent signals with intersecting group delay curves

TL;DR

The paper tackles the challenge of analyzing multicomponent signals with rapidly varying frequencies and intersecting group-delay () curves, where traditional time-frequency methods struggle to separate components. It develops a three-dimensional - representation based on the frequency-domain chirplet transform (FCT) and introduces the time-reassigned synchrosqueezing frequency-domain chirplet transform (TSFCT) together with a frequency-domain group signal separation operation (FGSSO) for accurate mode retrieval. The authors derive GD and reference functions with explicit error bounds and prove reconstruction error bounds for FGSSO, demonstrating sharp GD/GDD estimation and robust mode recovery in both simulated crossing-GD scenarios and real audio data. This framework yields sharper time-frequency representations and improved component separation for broadband transient signals, with practical implications for audio processing and bioacoustic analysis.

Abstract

To analyze signals with rapid frequency variations or transient components, the time-reassigned synchrosqueezing transform (TSST) and its variants have been recently proposed. Unlike the traditional synchrosqueezing transform, TSST squeezes the time-frequency (TF) coefficients along the group delay (GD) trajectories rather than the instantaneous frequency trajectories. Although TSST methods perform well in analyzing transient signals, they are fundamentally limited in processing multicomponent signals with intersecting GD curves. This limitation compromises the accuracy of both feature extraction and signal component recovery, thereby significantly reducing the interpretability of time-frequency representations (TFRs). This is particularly problematic in broadband signal processing systems, where the linearity of the phase response is critical and precise measurement of group delay dispersion (GDD) is essential. Motivated by the superior capability of frequency-domain signal modeling in characterizing rapidly frequency-varying signals, this paper proposes a novel three-dimensional time-frequency-group delay dispersion (TF-GDD) representation based on the frequency-domain chirplet transform. A subsequent time-reassigned synchrosqueezing frequency-domain chirplet transform (TSFCT) is introduced to achieve a sharper TF-GDD distribution and more accurate GD estimation. For mode retrieval, a novel frequency-domain group signal separation operation (FGSSO) is proposed.The theoretical contributions include a derivation of the approximation error for the GD and GDD reference functions and an establishment of the error bounds for FGSSO-based mode retrieval. Experimental results demonstrate that the proposed TSFCT and FGSSO effectively estimate GDs and retrieve modes--even for modes with intersecting GD trajectories.

Paper Structure

This paper contains 14 sections, 4 theorems, 90 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Time-reassigned synchrosqueezing frequency-domain chirplet transform
  3. Frequency-domain chirplet transform
  4. Time-reassigned synchrosqueezing frequency-domain chirplet transform
  5. Error analysis for GD and GDD reference functions
  6. Frequency-domain signal separation operation scheme
  7. Experimental results
  8. Implementation
  9. Numerical validation
  10. Audio recording of killer whale vocalizations
  11. Conclusion
  12. Expressions of the high-order GD and GDD reference functions
  13. Proof of Theorem \ref{['theorem_eta_lambda']}
  14. Error functions

Key Result

Theorem 1

Let $x(t) \in \mathcal{B}_{\epsilon_1, \epsilon_2}$ be a multicomponent signal with $K$ modes for sufficiently small $\epsilon_1, \epsilon_2 > 0$. Consider its FCT $\mathcal{D}_x^{\xi^m g}(t,\eta,\gamma)$ with window $\xi^m g(\xi)$. Suppose $(t, \eta, \gamma) \in Z_k$ and the matrix $E_0$ defined in where $\gamma_m$ and $\Lambda_{m,k}$ are defined in modulus_Cx and Lambdamk, respectively.

Figures (11)

  • Figure 1: GDs and GDDs of the signal $\widehat{x}(t)$
  • Figure 2: TFRs of $x(t)$. First row (from left to right): TSFCT, SST-2nd, TSST-2nd, TET-2nd; Second row: Corresponding local zoomed views of the TFRs in the first row.
  • Figure 3: GD and GDD estimation of $x(t)$ by TSFCT.
  • Figure 4: Recovery errors(real part) of $y(t)$ using the frequency-domain SSO algorithm. (a) Recovery error of $\widehat{x}_1(\eta)$; (b) recovery error of $\widehat{x}_2(\eta)$; (c) recovery error of $x_1(t)$; (d) recovery error of $x_2(t)$.
  • Figure 5: The infinity norm of the inverse coefficient matrix (left) and 2-norm condition number (right).
  • ...and 6 more figures

Theorems & Definitions (10)

  • Definition 1
  • Remark 2.1
  • Definition 2: Class $\mathcal{B}_{\epsilon_1,\epsilon_2}$
  • Theorem 1
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof