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A Unified Framework for 2D Nonseparable Fractional Fourier Transform: From Geometric Completeness to Applications

Daxiang Li, Zhichao Zhang, Wei Yao

Abstract

The one-dimensional (1D) fractional Fourier transform (FRFT) generalizes the Fourier transform, offering significant advantages in the time-frequency analysis of non-stationary signals. While various 2D extensions exist, such as the 2D separable FRFT (SFRFT), gyrator transform (GT), coupled FRFT (CFRFT), and earlier nonseparable definitions, they suffer from fragmented theoretical frameworks and a fundamental lack of geometric consistency with the 2D Wigner distribution (WD). Addressing these limitations, we propose a unified 2D nonseparable FRFT (NSFRFT) framework. Theoretically derived from the intersection of the symplectic and special orthogonal groups (isomorphic to the unitary group $\mathrm{U}(2)$), this transform inherently possesses four degrees of freedom and mathematically incorporates the 2D SFRFT, GT, and CFRFT as special cases. Unlike prior algebraic generalizations, it strictly preserves the rigid 4D rotational geometry of the 2D WD, ensuring geometric consistency and numerical stability. We derive its essential properties and develop efficient discrete algorithms with a computational complexity of $O(N^{2}\log N)$. Numerical simulations validate the superiority of the 2D NSFRFT in analyzing coupled chirp signals and demonstrate its robustness in filtering and image encryption and decryption applications.

A Unified Framework for 2D Nonseparable Fractional Fourier Transform: From Geometric Completeness to Applications

Abstract

The one-dimensional (1D) fractional Fourier transform (FRFT) generalizes the Fourier transform, offering significant advantages in the time-frequency analysis of non-stationary signals. While various 2D extensions exist, such as the 2D separable FRFT (SFRFT), gyrator transform (GT), coupled FRFT (CFRFT), and earlier nonseparable definitions, they suffer from fragmented theoretical frameworks and a fundamental lack of geometric consistency with the 2D Wigner distribution (WD). Addressing these limitations, we propose a unified 2D nonseparable FRFT (NSFRFT) framework. Theoretically derived from the intersection of the symplectic and special orthogonal groups (isomorphic to the unitary group ), this transform inherently possesses four degrees of freedom and mathematically incorporates the 2D SFRFT, GT, and CFRFT as special cases. Unlike prior algebraic generalizations, it strictly preserves the rigid 4D rotational geometry of the 2D WD, ensuring geometric consistency and numerical stability. We derive its essential properties and develop efficient discrete algorithms with a computational complexity of . Numerical simulations validate the superiority of the 2D NSFRFT in analyzing coupled chirp signals and demonstrate its robustness in filtering and image encryption and decryption applications.

Paper Structure

This paper contains 31 sections, 103 equations, 28 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The General Framework: 2D NSLCT
  4. Related 2D FRFTs
  5. Phase Space Analysis and Geometric Interpretation
  6. 4D Rotation Matrix Decomposition
  7. Definition, Properties, Geometric Interpretation and Distinctive Advantage of the 2D NSFRFT
  8. Definition of the 2D NSFRFT
  9. Properties of the 2D NSFRFT
  10. Geometric Interpretation of the 2D NSFRFT
  11. Unique Advantage: Handling Specific Coupled 2D Chirp Signals
  12. Discrete Algorithms of the 2D NSFRFT
  13. The Direct Method
  14. Fast Implementations via Orthogonal Symplectic Matrix Decomposition
  15. Comparisons between the Two Fast Algorithms and the Direct Method
  16. ...and 16 more sections

Figures (28)

  • Figure 1: The relationship between the 2D SFRFT,GT, and CFRFT. Here, $\alpha_1$, $\alpha_2$ are the parameters of the 2D SFRFT, $\varphi$ is the parameter of the GT, and $\alpha$, $\beta$ are the parameters of the CFRFT. The abbreviation "IT" denotes the 2D identity transform.
  • Figure 2: Hierarchical relationships between the 2D NSFRFT and its special cases.
  • Figure 3: 2D NSFRFT under the condition of input function g$_1$ and parameter matrix $\mathbf{X}_{\mathrm{ac1}}$. (a) Input function; (b) Output of the direct method; (c) Output of Algorithm I; (d) Output of Algorithm II.
  • Figure 4: 2D NSFRFT under the condition of input function g$_2$ and parameter matrix $\mathbf{X}_{\mathrm{ac1}}$. (a) Input function; (b) Output of the direct method; (c) Output of Algorithm I; (d) Output of Algorithm II.
  • Figure 5: 2D NSFRFT under the condition of input function g$_1$ and parameter matrix $\mathbf{X}_{\mathrm{ac2}}$. (a) Input function; (b) Output of the direct method; (c) Output of Algorithm I; (d) Output of Algorithm II.
  • ...and 23 more figures

Theorems & Definitions (10)

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