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Fractal Fract
Volume 7
Issue 2
10.3390/fractalfract7020159
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Article

An Interplay of Wigner–Ville Distribution and 2D Hyper-Complex Quadratic-Phase Fourier Transform

by
Mohammad Younus Bhat
1,
Aamir Hamid Dar
1,
Irfan Nurhidayat
2 and
Sandra Pinelas
3,*
1
Department of Mathematical Sciences, Islamic University of Science and Technology, Kashmir 192122, India
2
Department of Mathematics, School of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand
3
Departamento De Ciencias Exatas E Engenharia, Academia Militar, Av. Conde Castro Guimaraes, 2720-113 Amadora, Portugal
*
Author to whom correspondence should be addressed.
Fractal Fract. 2023, 7(2), 159; https://doi.org/10.3390/fractalfract7020159
Submission received: 5 January 2023 / Revised: 26 January 2023 / Accepted: 31 January 2023 / Published: 6 February 2023
(This article belongs to the Special Issue New Insights into Nonlinear Coupled Differential Equations with Its Applications)
Versions Notes

Abstract

Two-dimensional hyper-complex (Quaternion) quadratic-phase Fourier transforms (Q-QPFT) have gained much popularity in recent years because of their applications in many areas, including color image and signal processing. At the same time, the applications of Wigner–Ville distribution (WVD) in signal analysis and image processing cannot be ruled out. In this paper, we study the two-dimensional hyper-complex (Quaternion) Wigner–Ville distribution associated with the quadratic-phase Fourier transform (WVD-QQPFT) by employing the advantages of quaternion quadratic-phase Fourier transforms (Q-QPFT) and Wigner–Ville distribution (WVD). First, we propose the definition of the WVD-QQPFT and its relationship with the classical Wigner–Ville distribution in the quaternion setting. Next, we investigate the general properties of the newly defined WVD-QQPFT, including complex conjugate, symmetry-conjugation, nonlinearity, boundedness, reconstruction formula, Moyal’s formula, and Plancherel formula. Finally, we propose the convolution and correlation theorems associated with WVD-QQPFT.
Keywords: quaternion quadratic-phase Fourier transform; Winger–Ville distribution; boundedness; Moyals formula; convolution; correlation quaternion quadratic-phase Fourier transform; Winger–Ville distribution; boundedness; Moyals formula; convolution; correlation

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MDPI and ACS Style

Bhat, M.Y.; Dar, A.H.; Nurhidayat, I.; Pinelas, S. An Interplay of Wigner–Ville Distribution and 2D Hyper-Complex Quadratic-Phase Fourier Transform. Fractal Fract. 2023, 7, 159. https://doi.org/10.3390/fractalfract7020159

AMA Style

Bhat MY, Dar AH, Nurhidayat I, Pinelas S. An Interplay of Wigner–Ville Distribution and 2D Hyper-Complex Quadratic-Phase Fourier Transform. Fractal and Fractional. 2023; 7(2):159. https://doi.org/10.3390/fractalfract7020159

Chicago/Turabian Style

Bhat, Mohammad Younus, Aamir Hamid Dar, Irfan Nurhidayat, and Sandra Pinelas. 2023. "An Interplay of Wigner–Ville Distribution and 2D Hyper-Complex Quadratic-Phase Fourier Transform" Fractal and Fractional 7, no. 2: 159. https://doi.org/10.3390/fractalfract7020159

APA Style

Bhat, M. Y., Dar, A. H., Nurhidayat, I., & Pinelas, S. (2023). An Interplay of Wigner–Ville Distribution and 2D Hyper-Complex Quadratic-Phase Fourier Transform. Fractal and Fractional, 7(2), 159. https://doi.org/10.3390/fractalfract7020159

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