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Search Results (4)

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Keywords = quaternion quadratic-phase Fourier transform

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14 pages, 287 KB  
Article
Convolution, Correlation and Uncertainty Principle in the One-Dimensional Quaternion Quadratic-Phase Fourier Transform Domain
by Mohammad Younus Bhat, Aamir H. Dar, Mohra Zayed and Altaf A. Bhat
Mathematics 2023, 11(13), 3002; https://doi.org/10.3390/math11133002 - 5 Jul 2023
Cited by 5 | Viewed by 1286
Abstract
In this paper, we present a novel integral transform known as the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT). We first define the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT) of integrable (and square integrable) functions on R. Later on, we show that 1D-QQPFT [...] Read more.
In this paper, we present a novel integral transform known as the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT). We first define the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT) of integrable (and square integrable) functions on R. Later on, we show that 1D-QQPFT satisfies all the respective properties such as inversion formula, linearity, Moyal’s formula, convolution theorem, correlation theorem and uncertainty principle. Moreover, we use the proposed transform to obtain an inversion formula for two-dimensional quaternion quadratic-phase Fourier transform. Finally, we highlight our paper with some possible applications. Full article
(This article belongs to the Section E2: Control Theory and Mechanics)
15 pages, 302 KB  
Article
Some Essential Relations for the Quaternion Quadratic-Phase Fourier Transform
by Mawardi Bahri and Samsul Ariffin Abdul Karim
Mathematics 2023, 11(5), 1235; https://doi.org/10.3390/math11051235 - 3 Mar 2023
Cited by 8 | Viewed by 1616
Abstract
Motivated by the fact that the quaternion Fourier transform is a powerful tool in quaternion signal analysis, here, we study the quaternion quadratic-phase Fourier transform, which is a generalized version of the quaternion Fourier transform. We first give a definition of the quaternion [...] Read more.
Motivated by the fact that the quaternion Fourier transform is a powerful tool in quaternion signal analysis, here, we study the quaternion quadratic-phase Fourier transform, which is a generalized version of the quaternion Fourier transform. We first give a definition of the quaternion quadratic-phase Fourier transform. We derive in detail some essential properties related to this generalized transformation. We explore how the quaternion quadratic-phase Fourier transform is related to the quaternion Fourier transform. It is shown that this relation allows us to obtain several versions of uncertainty principles concerning the quaternion quadratic-phase Fourier transform. Full article
14 pages, 321 KB  
Article
An Interplay of Wigner–Ville Distribution and 2D Hyper-Complex Quadratic-Phase Fourier Transform
by Mohammad Younus Bhat, Aamir Hamid Dar, Irfan Nurhidayat and Sandra Pinelas
Fractal Fract. 2023, 7(2), 159; https://doi.org/10.3390/fractalfract7020159 - 6 Feb 2023
Cited by 9 | Viewed by 2161
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue New Insights into Nonlinear Coupled Differential Equations with Its Applications)
17 pages, 316 KB  
Article
Uncertainty Principles for the Two-Sided Quaternion Windowed Quadratic-Phase Fourier Transform
by Mohammad Younus Bhat, Aamir Hamid Dar, Irfan Nurhidayat and Sandra Pinelas
Symmetry 2022, 14(12), 2650; https://doi.org/10.3390/sym14122650 - 15 Dec 2022
Cited by 12 | Viewed by 1969
Abstract
A recent addition to the class of integral transforms is the quaternion quadratic-phase Fourier transform (Q-QPFT), which generalizes various signal and image processing tools. However, this transform is insufficient for addressing the quadratic-phase spectrum of non-stationary signals in the quaternion domain. To address [...] Read more.
A recent addition to the class of integral transforms is the quaternion quadratic-phase Fourier transform (Q-QPFT), which generalizes various signal and image processing tools. However, this transform is insufficient for addressing the quadratic-phase spectrum of non-stationary signals in the quaternion domain. To address this problem, we, in this paper, study the (two sided) quaternion windowed quadratic-phase Fourier transform (QWQPFT) and investigate the uncertainty principles associated with the QWQPFT. We first propose the definition of QWQPFT and establish its relation with quaternion Fourier transform (QFT); then, we investigate several properties of QWQPFT which includes inversion and the Plancherel theorem. Moreover, we study different kinds of uncertainty principles for QWQPFT such as Hardy’s uncertainty principle, Beurling’s uncertainty principle, Donoho–Stark’s uncertainty principle, the logarithmic uncertainty principle, the local uncertainty principle, and Pitt’s inequality. Full article
(This article belongs to the Special Issue Symmetry in Methods of Numerical Analysis and Its Application in Engineering)
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