Uncertainty principle for the two sided quaternion windowed Fourier transform

Brahim, Kamel; Tefjeni, Emna

doi:10.1007/s11868-019-00283-5

Cited by 28 publications

(15 citation statements)

References 16 publications

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“…, the Entropic uncertainty principle of the QWLCT becomes the Entropic uncertainty principle of the QWFT [48].…”

Section: Definitionmentioning

confidence: 99%

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Uncertainty principle for the two-sided quaternion windowed linear canonical transform

Gao¹,

Li²

2021

Preprint

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In this paper, we investigate the (two-sided) quaternion windowed linear canonical transform (QWLCT) and study the uncertainty principles associated with the QWLCT. Firstly, several important properties of the QWLCT such as bounded, shift, modulation, orthogonality relation, are presented based on the spectral representation of the quaternionic linear canonical transform (QLCT). Secondly, Pitt's inequality and Lieb inequality for the QWLCT are explored. Moreover, we study different kinds of uncertainty principles for the QWLCT, such as Logarithmic uncertainty principle, Entropic uncertainty principle, Lieb uncertainty principle and Donoho-Stark's uncertainty principle. Finally, we give a numerical example and a potential application in signal recovery by using Donoho-Stark's uncertainty principle associated with the QWLCT.

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“…, the Entropic uncertainty principle of the QWLCT becomes the Entropic uncertainty principle of the QWFT [48].…”

Section: Definitionmentioning

confidence: 99%

“…Moreover, the derived results in this paper can be extended to other transforms, such as Wigner Distribution (WD) and ambiguity function (AF) [35,48].…”

Section: Numerical Example and Potential Applicationmentioning

confidence: 99%

Uncertainty principle for the two-sided quaternion windowed linear canonical transform

Gao¹,

Li²

2021

Preprint

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“…We investigate several basic properties of the QWFT which are important for signal representation in signal processing. For more details on quaternion windowed Fourier transform, the reader can see [3,4,15,16,17].…”

Section: Corollarymentioning

confidence: 99%

“…The aim of this paper is to generalize the continuous quaternion windowed Fourier transform on R 2 to R 2d , called the multivariate two sided continuous quaternion windowed Fourier transform which has been started in [3,4]. Our purpose in this work is to prove three Lieb uncertainty principle for both of the QWFT, QWVT and QAF.…”

Section: Introductionmentioning

confidence: 99%

Uncertainty Principle for the multivariate two sided continuous quaternion Windowed Fourier transform

Brahim,

Tefjeni

2019

Preprint

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In this paper, we generalize the continuous quaternion windowed Fourier transform on R 2 to R 2d , called the multivariate two sided continuous quaternion windowed Fourier transform. Using the two sided quaternion Fourier transform, we derive several important properties such as (reconstruction formula, reproducing kernel, plancherel's formula, etc.). We present several example of the multivariate two sided continuous quaternion windowed Fourier transform. We apply the multivariate two sided continuous quaternion windowed Fourier transform properties and the two sided quaternion Fourier transform to establish Lieb uncertainty principle, the Logarithmic uncertainty principle the Beckner's uncertainty principle in term of entropy and the Heisenberg uncertainty principle for the multivariate two sided continuous quaternion windowed Fourier transform, the radar quaternion ambiguity function and the quaternion-Wigner transform. Last we study the multivariate two sided continuous quaternion windowed Fourier transform, the radar quaternion ambiguity function and the quaternion-Wigner transform on subset of finite measures.

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“…In this section, we recall some basic definitions and properties of the Quaternion Fourier transform. For more details, see [12,13,14,15]. The quaternion algebra was formally introduced by the Irish mathematician W.R Hamilton in 1843, and it is a generalization of complex numbers.…”

Section: Generalitiesmentioning

confidence: 99%

Lieb, Entropy and Logarithmic uncertainty principles for the multivariate continuous quaternion Shearlet Transform

Kamel,

Tefjeni,

Nefzi

2019

Preprint

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In this paper, we generalize the continuous quaternion shearlet transform on R 2 to R 2d , called the multivariate two sided continuous quaternion shearlet transform. Using the two sided quaternion Fourier transform, we derive several important properties such as (reconstruction formula, reproducing kernel, plancherel's formula, etc.). We present several example of the multivariate two sided continuous quaternion shearlet transform. We apply the multivariate two sided continuous quaternion shearlet transform properties and the two sided quaternion Fourier transform to establish Lieb uncertainty principle and the Logarithmic uncertainty principle. Last we study the Beckner's uncertainty principle in term of entropy for the multivariate two sided continuous quaternion shearlet transform.

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Uncertainty principle for the two sided quaternion windowed Fourier transform

Cited by 28 publications

References 16 publications

Uncertainty principle for the two-sided quaternion windowed linear canonical transform

Uncertainty principle for the two-sided quaternion windowed linear canonical transform

Uncertainty Principle for the multivariate two sided continuous quaternion Windowed Fourier transform

Lieb, Entropy and Logarithmic uncertainty principles for the multivariate continuous quaternion Shearlet Transform

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