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Graph Linear Canonical Transform Based on CM-CC-CM Decomposition

Na Li, Zhichao Zhang, Jie Han, Yunjie Chen, Chunzheng Cao

TL;DR

This paper introduces a GLCT based on chirp multiplication-chirp convolution-chirp multiplication decomposition (CM-CC-CM-GLCT), which demonstrates that the computational complexity of the CM-CC-CM-GLCT is significantly reduced.

Abstract

The graph linear canonical transform (GLCT) is presented as an extension of the graph Fourier transform (GFT) and the graph fractional Fourier transform (GFrFT), offering more flexibility as an effective tool for graph signal processing. In this paper, we introduce a GLCT based on chirp multiplication-chirp convolution-chirp multiplication decomposition (CM-CC-CM-GLCT), which irrelevant to sampling periods and without oversampling operation. Various properties and special cases of the CM-CC-CM-GLCT are derived and discussed. In terms of computational complexity, additivity, and reversibility, we compare the CM-CC-CM-GLCT and the GLCT based on the central discrete dilated Hermite function (CDDHFs-GLCT). Theoretical analysis demonstrates that the computational complexity of the CM-CC-CM-GLCT is significantly reduced. Simulation results indicate that the CM-CC-CM-GLCT achieves similar additivity to the CDDHFs-GLCT. Notably, the CM-CC-CM-GLCT exhibits better reversibility.

Graph Linear Canonical Transform Based on CM-CC-CM Decomposition

TL;DR

This paper introduces a GLCT based on chirp multiplication-chirp convolution-chirp multiplication decomposition (CM-CC-CM-GLCT), which demonstrates that the computational complexity of the CM-CC-CM-GLCT is significantly reduced.

Abstract

The graph linear canonical transform (GLCT) is presented as an extension of the graph Fourier transform (GFT) and the graph fractional Fourier transform (GFrFT), offering more flexibility as an effective tool for graph signal processing. In this paper, we introduce a GLCT based on chirp multiplication-chirp convolution-chirp multiplication decomposition (CM-CC-CM-GLCT), which irrelevant to sampling periods and without oversampling operation. Various properties and special cases of the CM-CC-CM-GLCT are derived and discussed. In terms of computational complexity, additivity, and reversibility, we compare the CM-CC-CM-GLCT and the GLCT based on the central discrete dilated Hermite function (CDDHFs-GLCT). Theoretical analysis demonstrates that the computational complexity of the CM-CC-CM-GLCT is significantly reduced. Simulation results indicate that the CM-CC-CM-GLCT achieves similar additivity to the CDDHFs-GLCT. Notably, the CM-CC-CM-GLCT exhibits better reversibility.
Paper Structure (13 sections, 40 equations, 6 figures, 4 tables)

This paper contains 13 sections, 40 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Review of CDDHFs-GLCT
  4. CM-CC-CM-GLCT
  5. Formulation of the CM-CC-CM-GLCT for $b\ne 0$
  6. Formulation of the CM-CC-CM-GLCT for $b=0$
  7. Properties
  8. Special Cases
  9. Comparison between CM-CC-CM-GLCT and CDDHFS-GLCT
  10. Computational complexity
  11. Additivity property
  12. Revertibility property
  13. Conclusions

Figures (6)

  • Figure 1: Graph signals $\mathbf{x}_{1}$ to $\mathbf{x}_{8}$.
  • Figure 2: Normalized mean-square errors (NMSEs) of the additivity property for 50 different sets of $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$.
  • Figure 3: Normalized mean-square errors (NMSEs) of the reversibility property for 50 different sets of $\mathbf{M}$.
  • Figure :
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  • ...and 1 more figures