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Symplectic Wigner Distribution in the Linear Canonical Transform Domain: Theory and Application

Yangfan He, Zhichao Zhang

TL;DR

The paper presents the SWDL, a unifying Wigner distribution in the LCT domain that merges symplectic (SWD) and chirp-based (WDL) approaches. It establishes a comprehensive theoretical framework, including definitions, equivalent LCT representations, and a full set of fundamental properties, plus Heisenberg-type uncertainty principles and lower bounds. A symplectic-matrix optimization strategy is developed to maximize time-frequency resolution, with rigorous results showing SWDL outperforms SWD and WDL for LFM signals. Numerical experiments on an LFM example confirm enhanced energy concentration and practical feasibility, underscoring SWDL’s potential for radar, communications, and related applications.

Abstract

This paper devotes to combine the chirp basis function transformation and symplectic coordinates transformation to yield a novel Wigner distribution (WD) associated with the linear canonical transform (LCT), named as the symplectic WD in the LCT domain (SWDL). It incorporates the merits of the symplectic WD (SWD) and the WD in the LCT domain (WDL), achieving stronger capability in the linear frequency-modulated (LFM) signal frequency rate feature extraction while maintaining the same level of computational complexity. Some essential properties of the SWDL are derived, including marginal distributions, energy conservations, unique reconstruction, Moyal formula, complex conjugate symmetry, time reversal symmetry, scaling property, time translation property, frequency modulation property, and time translation and frequency modulation property. Heisenberg's uncertainty principles of the SWDL are formulated, giving rise to three kinds of lower bounds attainable respectively by Gaussian enveloped complex exponential signal, Gaussian signal and Gaussian enveloped chirp signal. The optimal symplectic matrices corresponding to the highest time-frequency resolution are generated by solving the lower bound optimization (minimization) problem. The time-frequency resolution of the SWDL is compared with those of the SWD and WDL to demonstrate its superiority in LFM signals time-frequency energy concentration. A synthesis example is also carried out to verify the feasibility and reliability of the theoretical analysis.

Symplectic Wigner Distribution in the Linear Canonical Transform Domain: Theory and Application

TL;DR

The paper presents the SWDL, a unifying Wigner distribution in the LCT domain that merges symplectic (SWD) and chirp-based (WDL) approaches. It establishes a comprehensive theoretical framework, including definitions, equivalent LCT representations, and a full set of fundamental properties, plus Heisenberg-type uncertainty principles and lower bounds. A symplectic-matrix optimization strategy is developed to maximize time-frequency resolution, with rigorous results showing SWDL outperforms SWD and WDL for LFM signals. Numerical experiments on an LFM example confirm enhanced energy concentration and practical feasibility, underscoring SWDL’s potential for radar, communications, and related applications.

Abstract

This paper devotes to combine the chirp basis function transformation and symplectic coordinates transformation to yield a novel Wigner distribution (WD) associated with the linear canonical transform (LCT), named as the symplectic WD in the LCT domain (SWDL). It incorporates the merits of the symplectic WD (SWD) and the WD in the LCT domain (WDL), achieving stronger capability in the linear frequency-modulated (LFM) signal frequency rate feature extraction while maintaining the same level of computational complexity. Some essential properties of the SWDL are derived, including marginal distributions, energy conservations, unique reconstruction, Moyal formula, complex conjugate symmetry, time reversal symmetry, scaling property, time translation property, frequency modulation property, and time translation and frequency modulation property. Heisenberg's uncertainty principles of the SWDL are formulated, giving rise to three kinds of lower bounds attainable respectively by Gaussian enveloped complex exponential signal, Gaussian signal and Gaussian enveloped chirp signal. The optimal symplectic matrices corresponding to the highest time-frequency resolution are generated by solving the lower bound optimization (minimization) problem. The time-frequency resolution of the SWDL is compared with those of the SWD and WDL to demonstrate its superiority in LFM signals time-frequency energy concentration. A synthesis example is also carried out to verify the feasibility and reliability of the theoretical analysis.

Paper Structure

This paper contains 19 sections, 59 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metaplectic representation
  4. Symplectic WD (SWD)
  5. WD in the LCT domain (WDL)
  6. SWD in the LCT domain (SWDL)
  7. Definition
  8. Definition revisited in terms of the LCT
  9. Main properties
  10. Computational complexity
  11. Heisenberg's uncertainty principles in the SWDL setting
  12. Uncertainty product of the SWDL
  13. Lower bounds on the uncertainty product
  14. Symplectic matrices selection strategy for LFM signal processing
  15. The optimal symplectic matrices
  16. ...and 4 more sections

Figures (1)

  • Figure 1: The SWDL, SWD, WDL and WD of the LFM signal. (a) Contour picture of the SWDL; (b) Frequency rate-amplitude distribution of the RT-SWDL; (c) Contour picture of the SWD; (d) Frequency rate-amplitude distribution of the RT-SWD; (e) Contour picture of the WDL; (f) Frequency rate-amplitude distribution of the RT-WDL; (g) Contour picture of the WD; (h) Frequency rate-amplitude distribution of the RT-WD.