Uncertainty principle for the two-sided quaternion windowed Fourier transform

Brahim, Kamel; Tefjeni, Emna

doi:10.1080/10652469.2019.1572138

Cited by 18 publications

(3 citation statements)

References 12 publications

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“…As the classical Fourier transform, the QFT is ineffective in representing and computing local information of signals. To overcome this drawback, the windowed quaternionic Fourier transform (WQFT) and quaternionic Gabor systems were introduced and studied [1, 2, 6, 11, 15, 18‐20, 24, 26, 35]. Given

\mathbf{a}

\mathbf{b}\in {\mathrm{\mathbb{R}}}&#x0005E;2

and

\left({\eta}_1,{\eta}_2\right)\in \left\{1,i\right\}\times \left\{1,j\right\}

, define the translation operator

{T}_{\mathbf{a}}

and modulation operator

{M}_{\mathbf{b}}&#x0005E;{\left({\eta}_1,{\eta}_2\right)}

{L}&#x0005E;2\left({\mathrm{\mathbb{R}}}&#x0005E;2,\mathrm{\mathbb{H}}\right)

{T}_{\mathbf{a}}f\left(\mathbf{x}\right)&#x0003D;f\left(\mathbf{x}-\mathbf{a}\right),

and

M_{boldb}^{false(η_{1}, η_{2} false)} f false(boldx false) = η_{2} e^{2 π}

…”

Section: Introductionmentioning

confidence: 99%

Characterization of rationally sampled quaternionic dual Gabor frames

Li,

Zhang

2023

Math Methods in App Sciences

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This paper addresses quaternionic dual Gabor frames under the condition that the products of time‐frequency shift parameters are rational numbers. For a general overcomplete quaternionic Gabor frame with the product of time‐frequency shift parameters not equal to , we show that its corresponding frame and translation operators do not commute, which leads to its canonical dual frame not having the Gabor structure, but it may have other dual frames with Gabor structure. We characterize when two quaternionic Gabor Bessel sequences form a pair of dual frames, and present a class of quaternionic dual Gabor frames. We also characterize quaternionic Gabor Riesz bases and prove that their canonical dual frames have Gabor structure.

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\mathbf{a}

\mathbf{b}\in {\mathrm{\mathbb{R}}}&#x0005E;2

and

\left({\eta}_1,{\eta}_2\right)\in \left\{1,i\right\}\times \left\{1,j\right\}

, define the translation operator

{T}_{\mathbf{a}}

and modulation operator

{M}_{\mathbf{b}}&#x0005E;{\left({\eta}_1,{\eta}_2\right)}

{L}&#x0005E;2\left({\mathrm{\mathbb{R}}}&#x0005E;2,\mathrm{\mathbb{H}}\right)

{T}_{\mathbf{a}}f\left(\mathbf{x}\right)&#x0003D;f\left(\mathbf{x}-\mathbf{a}\right),

and

M_{boldb}^{false(η_{1}, η_{2} false)} f false(boldx false) = η_{2} e^{2 π}

…”

Section: Introductionmentioning

confidence: 99%

Characterization of rationally sampled quaternionic dual Gabor frames

Li,

Zhang

2023

Math Methods in App Sciences

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show abstract

“…They also studied the Pitt's inequality, Lieb's inequality and the logarithmic UP for the two sided QWFT studied in [5]. Including, the orthogonality property, authors in [24], [8] studied the local UP, logarithmic UP, Beckner's UP in terms of entropy, Lieb's UP, Amrein-Berthier UP for the two sided QWFT. Replacing the Fourier kernel in the left sided, right sided or two sided QWFT by the kernels of the FrFT (or LCT), results in the left sided, right sided and two sided QWFrFT (or QWLCT) respectively.…”

Section: Introductionmentioning

confidence: 99%

Short time quaternion quadratic phase Fourier transform and its uncertainty principles

Gupta¹,

Verma²

2022

Preprint

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In this paper, we extend the quadratic phase Fourier transform of a complex valued functions to that of the quaternion valued functions of two variables. We call it the quaternion quadratic phase Fourier transform (QQPFT). Based on the relation between the QQPFT and the quaternion Fourier transform (QFT) we obtain the sharp Hausdorff-Young inequality for QQPFT. We define the short time quaternion quadratic phase Fourier transform (STQQPFT) and explore some of its properties including inner product relation and inversion formula. We find its relation with that of the 2D quaternion ambiguity function and the quaternion Wigner-Ville distribution associated with QQPFT and obtain the Lieb's uncertainty and entropy uncertainty principles for these three transforms.

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“…e quaternion Fourier transform (QFT) is a nontrivial generalization of the classical Fourier transform (FT) using the quaternion algebra. e QFT has been shown to relate to the other quaternion signal analysis tools, such as quaternion wavelet transform [1][2][3], fractional quaternion Fourier transform [4,5], quaternionic windowed Fourier transform [6][7][8][9], and quaternion Wigner transform [10]. Because of the noncommutative property of quaternion multiplication, we obtain at least three different kinds of two-dimensional QFTs as follows (see [11][12][13][14][15]):…”

Section: Introductionmentioning

confidence: 99%

Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle

Bahri

Ashino

2019

Journal of Mathematics

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A definition of the two-dimensional quaternion linear canonical transform (QLCT) is proposed. The transform is constructed by substituting the Fourier transform kernel with the quaternion Fourier transform (QFT) kernel in the definition of the classical linear canonical transform (LCT). Several useful properties of the QLCT are obtained from the properties of the QLCT kernel. Based on the convolutions and correlations of the LCT and QFT, convolution and correlation theorems associated with the QLCT are studied. An uncertainty principle for the QLCT is established. It is shown that the localization of a quaternion-valued function and the localization of the QLCT are inversely proportional and that only modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.

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Uncertainty principle for the two-sided quaternion windowed Fourier transform

Cited by 18 publications

References 12 publications

Characterization of rationally sampled quaternionic dual Gabor frames

Characterization of rationally sampled quaternionic dual Gabor frames

Short time quaternion quadratic phase Fourier transform and its uncertainty principles

Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle

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