Synchrosqueezed windowed linear canonical transform: A method for mode retrieval from multicomponent signals with crossing instantaneous frequencies

TL;DR

This work tackles the problem of separating multicomponent, non-stationary signals with crossing instantaneous frequencies by developing a windowed linear canonical transform (WLCT) that elevates the time-frequency plane to a three-dimensional time-frequency-chirprate space. It introduces four WLCT variants and an X-ray WLCT (XWLCT), and then derives two synchrosqueezed formulations (SWLCT and SXWLCT) to sharpen the representation and enable robust mode retrieval via ridge extraction and a reconstruction scheme under separability conditions. The approach includes entropy-guided parameter selection to maximize concentration, and demonstrates superior or competitive IF and chirprate estimation compared to existing methods like TET and MESCT across synthetic and real signals, with publicly available code. Collectively, the WLCT framework and its synchrosqueezed extensions provide a flexible, three-dimensional toolbox for accurate mode separation in challenging multicomponent signals.

Abstract

In nature, signals often appear in the form of the superposition of multiple non-stationary signals. The overlap of signal components in the time-frequency domain poses a significant challenge for signal analysis. One approach to addressing this problem is to introduce an additional chirprate parameter and use the chirplet transform (CT) to elevate the two-dimensional time-frequency representation to a three-dimensional time-frequency-chirprate representation. From a certain point of view, the CT of a signal can be regarded as a windowed special linear canonical transform of that signal, undergoing a shift and a modulation. In this paper, we develop this idea to propose a novel windowed linear canonical transform (WLCT), which provides a new time-frequency-chirprate representation. We discuss four types of WLCTs. In addition, we use a special X-ray transform to further sharpen the time-frequency-chirprate representation. Furthermore, we derive the corresponding three-dimensional synchrosqueezed transform, demonstrating that the WLCTs have great potential for three-dimensional signal separation.

Figures (16)

• Figure 1: Time-frequency representations of $x(t)$. (a) IFs of signal; (b) SST; (c) 2nd SST; (d) MSST; (e) STFT; (f) SST (local zoom); (g) 2nd SST (local zoom); (h) MSST (local zoom).
• Figure 2: IF and chirprate estimations, and real part errors of mode retrieval: First row by $S^1_x(t, \xi, \gamma)$, Second row by $S^5_x(t, \xi, \gamma)$, Third row by $S^2_x(t, \xi, \gamma)$ and Fourth row by $S^6_x(t, \xi, \gamma)$.
• Figure 3: (a) reconstruction error with estimated IFs and chirprates; (b) condition numbers of inverse coefficient matrices.
• Figure 4: (a) IFs with small perturbations; (b) chirprates with small perturbations; (c) Reconstruction errors with small perturbations.
• Figure 5: IFs and chirprates of $y_1, y_2$. (a) IFs; (b) Chirprates.

Theorems & Definitions (6)

• Definition 1
• Definition 2
• Remark 3.1
• Definition 3
• Definition 4
• Definition 5