scholarly journals Relation between Quaternion Fourier Transform and Quaternion Wigner-Ville Distribution Associated with Linear Canonical Transform

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Mawardi Bahri ◽  
Muh. Saleh Arif Fatimah
Keyword(s):  
Fourier Transform ◽  
Inversion Formula ◽  
Ambiguity Function ◽  
Alternative Proof ◽  
Basic Relation ◽  
Linear Canonical Transform ◽  
Quaternion Fourier Transform ◽  
Fundamental Relationship ◽  
The Relationship

The quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) is a nontrivial generalization of the quaternion Wigner-Ville distribution to the linear canonical transform (LCT) domain. In the present paper, we establish a fundamental relationship between the QWVD-LCT and the quaternion Fourier transform (QFT). Based on this fact, we provide alternative proof of the well-known properties of the QWVD-LCT such as inversion formula and Moyal formula. We also discuss in detail the relationship among the QWVD-LCT and other generalized transforms. Finally, based on the basic relation between the quaternion ambiguity function associated with the linear canonical transform (QAF-LCT) and the QFT, we present some important properties of the QAF-LCT.

scholarly journals A Version of Uncertainty Principle for Quaternion Linear Canonical Transform

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Mawardi Bahri ◽  
Resnawati ◽  
Selvy Musdalifah
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Young Inequality ◽  
Linear Canonical Transform ◽  
Two Dimensional ◽  
Research Papers ◽  
Quaternion Fourier Transform ◽  
Quaternion Linear Canonical Transform ◽  
The Relationship

In recent years, the two-dimensional (2D) quaternion Fourier and quaternion linear canonical transforms have been the focus of many research papers. In the present paper, based on the relationship between the quaternion Fourier transform (QFT) and the quaternion linear canonical transform (QLCT), we derive a version of the uncertainty principle associated with the QLCT. We also discuss the generalization of the Hausdorff-Young inequality in the QLCT domain.

scholarly journals Modifikasi Teorema Konvolusi Transformasi Kanonikal Linier Quaternion Sisi Kiri

2020 ◽  
Vol 16 (2) ◽  
pp. 116-125
Author(s):  
W Oktaviana ◽  
S Musdalifah ◽  
Resnawati
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Convolution Theorem ◽  
Linear Canonical Transform ◽  
Quaternion Fourier Transform ◽  
Fundamental Relationship ◽  
Quaternion Linear Canonical Transform

ABSTRACTQuaternion Linear Canonical Transform (QLCT) is a generalization of the Linear Canonical Transform (LCT) onquaternion algebra which plays an important role in optics and signal processing. QLCT can be seen asgeneralization of Quaternion Fourier Transform (QFT). Based on the fact, this paper propose the formulat ofmodification convolution theorem based on left-sided QLCT by considering the fundamental relationship betweenleft-sided QLCT and QFT. The results showed the modification convolution theorem for left-sided QLCT based onrelationship between left-sided QLCT and QFT as a sum of multiplication of left-sided QFT can be derived.Keywords : Convolution Theorem, Quaternion Fourier Transform, Quaternion Linear Canonical Transform

scholarly journals A Simplified Proof of Uncertainty Principle for Quaternion Linear Canonical Transform

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Mawardi Bahri ◽  
Ryuichi Ashino
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Simple Proof ◽  
Linear Canonical Transform ◽  
Quaternion Fourier Transform ◽  
Inverse Transform ◽  
Fundamental Relationship ◽  
Quaternion Linear Canonical Transform

We provide a short and simple proof of an uncertainty principle associated with the quaternion linear canonical transform (QLCT) by considering the fundamental relationship between the QLCT and the quaternion Fourier transform (QFT). We show how this relation allows us to derive the inverse transform and Parseval and Plancherel formulas associated with the QLCT. Some other properties of the QLCT are also studied.

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

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Mawardi Bahri ◽  
Ryuichi Ashino
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Linear Canonical Transform ◽  
Two Dimensional ◽  
Quaternion Fourier Transform ◽  
Gaussian Functions ◽  
The Fourier Transform ◽  
Definition Of ◽  
Quaternion Linear Canonical Transform

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.

scholarly journals The Generalization of the Poisson Sum Formula Associated with the Linear Canonical Transform

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Jun-Fang Zhang ◽  
Shou-Ping Hou
Keyword(s):  
Fourier Transform ◽  
Canonical Transformation ◽  
Linear Canonical Transform ◽  
Original Signal ◽  
Periodic Property ◽  
Sum Formula ◽  
The Relationship

The generalization of the classical Poisson sum formula, by replacing the ordinary Fourier transform by the canonical transformation, has been derived in the linear canonical transform sense. Firstly, a new sum formula of Chirp-periodic property has been introduced, and then the relationship between this new sum and the original signal is derived. Secondly, the generalization of the classical Poisson sum formula to the linear canonical transform sense has been obtained.

Uncertainty principles related to quaternionic windowed Fourier transform

2020 ◽  
Vol 18 (03) ◽  
pp. 2050015
Author(s):  
Mawardi Bahri ◽  
Ryuichi Ashino
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principles ◽  
Quaternion Fourier Transform ◽  
Windowed Fourier Transform ◽  
The Relationship

In this paper, we first introduce uncertainty principles for the quaternion Fourier transform (QFT). We then provide a different proof of the well-known properties of the quaternionic windowed Fourier transform (QWFT) using properties of the QFT which is a little bit simpler than usual. Based on uncertainty principles for the QFT and the relationship between the QFT and QWFT, we establish uncertainty principles related to the QWFT.

scholarly journals Two-sided fractional quaternion Fourier transform and its application

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zunfeng Li ◽  
Haipan Shi ◽  
Yuying Qiao
Keyword(s):  
Fourier Transform ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential ◽  
Quaternion Fourier Transform ◽  
Inverse Transform ◽  
Definition Of ◽  
Differential Properties ◽  
The Relationship

AbstractIn this paper, we introduce the two-sided fractional quaternion Fourier transform (FrQFT) and give some properties of it. The main results of this paper are divided into three parts. Firstly we give a definition of the FrQFT. Secondly based on properties of the two-sided QFT, we study the relationship between the two-sided QFT and the two-sided FrQFT, and give some differential properties of the two-sided FrQFT and the Parseval identity. Finally, we give an example to illustrate the application of the two-sided FrQFT and its inverse transform in solving partial differential equations.

Wavelet packets associated with linear canonical transform on spectrum

2021 ◽  
pp. 2150030
Author(s):  
M. Younus Bhat ◽  
Aamir H. Dar
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Integral Transforms ◽  
Wavelet Packets ◽  
Linear Canonical Transform ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Vector Valued

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.

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