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convolution theorem for the windowed linear canonical transform

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 2025年第2期36卷 91-101页

作者： Gao, Wen-Biao Yangzhou Univ Sch Math Sci Yangzhou 225002 Peoples R China

In this paper, we obtain the convolution theorems for the windowed linear canonical transform (WLCT). According to the WLCT of a convolution of two functions is the product of their respective WLCTs, the spectral convolution theorem of the WLCT is derived. Then, the spatial convolution theorem of the WLCT is exploited by a new spatial convolution operator. Moreover, applying the mathematical inequalities, the existence theorems of the convolution for the WLCT are established. Finally, as an application, the solution of a convolution equation is given by the convolution theorem of the WLCT.

关键词： Linear canonical transform windowed linear canonical transform convolution theorem convolution equation

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Holographic Lens Resolution Using the convolution theorem

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POLYMERS 2022年第24期14卷 5426页

作者： Lloret, Tomas Morales-Vidal, Marta Navarro-Fuster, Victor Ramirez, Manuel G. Belendez, Augusto Pascual, Inmaculada Univ Alicante Dept Opt Farmacol & Anat Carretera San Vicente Raspeig S-N San Vicente Del Raspeig 03690 Spain Univ Alicante Inst Univ Fis Aplicada Ciencias & Tecnol Carretera San Vicente Raspeig S-N San Vicente Del Raspeig 03690 Spain Univ Alicante Dept Fis Ingn Sistemas & Teoria Senal Carretera San Vicente Raspeig S-N San Vicente Del Raspeig 03690 Spain

The similarity between object and image of negative asymmetrical holographic lenses (HLs) stored in a low-toxicity photopolymer has been evaluated theoretically and experimentally. Asymmetrical experimental setups with negative focal lengths have been used to obtain HLs. For this purpose, the resolution of the HLs was calculated using the convolution theorem. A USAF 1951 test was used as an object and the impulse responses of the HLs, which in this case was the amplitude spread function (ASF), were obtained with two different methods: using a CCD sensor and a Hartmann Shack (HS) wavefront sensor. For a negative asymmetrically recorded HL a maximum resolution of 11.31 lp/mm was obtained. It was evaluated at 473 nm wavelength. A theoretical study of object-image similarity had carried out using the MSE (mean squared error) metric to evaluate the experimental results obtained quantitatively.

关键词： holographic lenses resolution convolution theorem volume holography

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convolution theorem with Its Derivatives and Multiresolution Analysis for Fractional S-Transform

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CIRCUITS SYSTEMS AND SIGNAL PROCESSING 2019年第11期38卷 5212-5235页

作者： Ranjan, Rajeev Singh, A. K. Jindal, Neeru Thapar Inst Engn & Technol Dept Elect & Commun Engn Patiala Punjab India

Fractional S-transform (FrST) is a time-frequency representation of signals with frequency-dependent resolution. FrST is also an advantageous technique for non-stationary signal processing applications. Till now, only linearity, scaling, time reversal, time marginal condition, and inverse FrST properties are documented. In this paper, some remaining properties of FrST are proposed to establish it as a complete transform technique. The proposed properties are convolution theorem, correlation theorem, and Parseval's theorem. To expand the applicability of FrST as a mathematical transform tool, the multiresolution analysis concept is also documented. The multiresolution analysis has shown significant performance to develop the orthogonal kernel for FrST. Finally, the applications of proposed convolution theorem are demonstrated on multiplicative filtering for electrocardiogram signal and linear frequency-modulated signal under AWGN channel.

关键词： Fractional S-transform convolution theorem Correlation theorem Parseval's theorem Multiresolution analysis Multiplicative filtering

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A new convolution theorem associated with the linear canonical transform

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SIGNAL IMAGE AND VIDEO PROCESSING 2019年第1期13卷 127-133页

作者： Huo, Haiye Nanchang Univ Sch Sci Dept Math Nanchang 330031 Jiangxi Peoples R China

In this paper, we first introduce a new notion of canonical convolution operator, and show that it satisfies the commutative, associative, and distributive properties, which may be quite useful in signal processing. Moreover, it is proved that the generalized convolution theorem and generalized Young's inequality are also hold for the new canonical convolution operator associated with the LCT. Finally, we investigate the sufficient and necessary conditions for solving a class of convolution equations associated with the LCT.

关键词： convolution operator convolution theorem Linear canonical transform Young's inequality convolution equations

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Fractional convolution, correlation theorem and its application in filter design

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SIGNAL IMAGE AND VIDEO PROCESSING 2020年第2期14卷 351-358页

作者： Feng, Qiang Wang, Rong-Bo Yanan Univ Sch Math & Comp Sci Yanan 716000 Shaanxi Peoples R China

In this paper, fractional convolution and correlation structures are proposed. The corresponding theorems for fractional Fourier transform (FRFT) are derived, which state that fractional convolution in the time domain is equivalent to a simple multiplication operation for FRFT and FT domain;this feature is more instrumental for the multiplicative filter model in FRFT domain. Moreover, the fractional convolution operation proposed in this paper can be expressed as ordinary convolution form in FT domain;such expression is particularly useful and easy to implement in filter design in time domain. Classical convolution and correlation theorems for Fourier transform (FT) are shown to be special case of our achieved result. The potential application of the derived result in filter design is also discussed, and the proposed method has lower computational complexity and can be easily implemented in FRFT domain.

关键词： Fractional Fourier transform Fractional convolution Fractional correlation convolution theorem Filter design

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convolution theorem for fractional cosine-sine transform and its application

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2017年第10期40卷 3651-3665页

作者： Feng, Qiang Li, Bing-Zhao Beijing Inst Technol Sch Math & Stat Beijing 102488 Peoples R China Beijing Inst Technol Beijing Key Lab Math Characterizat Anal & Applica Beijing 102488 Peoples R China Yanan Univ Sch Math & Comp Sci Yanan 716000 Shaanxi Peoples R China

Fractional cosine transform (FRCT) and fractional sine transform (FRST), which are closely related to the fractional Fourier transform (FRFT), are useful mathematical and optical tool for signal processing. Many properties for these transforms are well investigated, but the convolution theorems are still to be determined. In this paper, we derive convolution theorems for the fractional cosine transform (FRCT) and fractional sine transform (FRST) based on the four novel convolution operations. And then, a potential application for these two transforms on designing multiplicative filter is presented. Copyright (c) 2016 John Wiley & Sons, Ltd.

关键词： Fractional Fourier transform fractional cosine and sine transform convolution operation convolution theorem

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Linear Canonical Space-Time Transform and convolution theorems

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ADVANCES IN APPLIED CLIFFORD ALGEBRAS 2025年第3期35卷 1-18页

作者： Xu, Yi-Qiao Li, Bing-Zhao Beijing Inst Technol Sch Math & Stat Beijing 100081 Peoples R China Beijing Inst Technol Beijing Key Lab MCAACI Beijing 100081 Peoples R China

Following the idea of the fractional space-time Fourier transform, a linear canonical space-time transform for 16-dimensional space-time C & ell;3,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\ell _{3,1}$$\end{document}-valued signals is investigated in this paper. First, the definition of the proposed linear canonical space-time transform is given, and some related properties of this transform are obtained. Second, the convolution operator and the corresponding convolution theorem are proposed. Third, the convolution theorem associated with the two-sided linear canonical space-time transform is derived.

关键词： Space-time algebra Fractional space-time Fourier transform Linear canonical transform Linear canonical space-time transform convolution theorem

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convolution theorems associated with quaternion linear canonical transform and applications

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SIGNAL PROCESSING 2023年 202卷

作者： Hu, Xiaoxiao Cheng, Dong Kou, Kit Ian Wenzhou Med Univ Affiliated Hosp 1 Wenzhou Zhejiang Peoples R China Beijing Normal Univ Zhuhai Res Ctr Math & Math Educ Zhuhai Peoples R China Univ Macau Fac Sci & Technol Dept Math Macau Peoples R China

Novel types of convolution operators for quaternion linear canonical transform (QLCT) are proposed. Type one and two are defined in the spatial and QLCT spectral domains, respectively. They are distinct in the quaternion space and are consistent once in complex or real space. Various types of convolution for-mulas are discussed. Consequently, the QLCT of the convolution of two quaternionic functions can be implemented by the product of their QLCTs, or the summation of the products of their QLCTs. As appli-cations, correlation operators and theorems of the QLCT are derived. The proposed convolution formulas are used to solve Fredholm integral equations with special kernels. Some systems of second-order par-tial differential equations, which can be transformed into the second-order quaternion partial differential equations, can be solved by the convolution formulas as well. As a final point, we demonstrate that the convolution theorem facilitates the design of multiplicative filters.(c) 2022 Elsevier B.V. All rights reserved.

关键词： Quaternion linear canonical transform convolution theorem Fredholm integral equation Quaternion partial differential equations Multiplication filters

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The explicit solutions for a class of fractional Fourier-Laplace convolution equations

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INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 2023年第2期34卷 128-144页

作者： Feng, Q. Yuan, S. Yanan Univ Sch Math & Comp Sci Yanan 716000 Shaanxi Peoples R China

In this paper, two types of fractional Fourier-Laplace convolutions are defined, and the corresponding fractional Fourier-Laplace convolution theorems associated with the fractional cosine transform (FRCT), fractional sine transform (FRST) and Laplace transform (LP) are investigated in detail. The relationship between the fractional Fourier-Laplace convolutions and the existing convolutions is given, and Young's type theorem as well as the weighted convolution inequality are also obtained. As an application for fractional Fourier-Laplace convolution, the filter design and the system of convolution-type integral equations are also considered, the computational complexity for the multiplicative filter is analysed, and explicit solutions for these equations are obtained.

关键词： Fractional Fourier transform Laplace transform convolution theorem filter convolution equation

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The octonion linear canonical transform: Properties and applications

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CHAOS SOLITONS & FRACTALS 2025年 192卷

作者： Jiang, Nan Feng, Qiang Yang, Xi He, Jin-Rong Li, Bing-Zhao Yanan Univ Sch Math & Comp Sci Yanan 716000 Shaanxi Peoples R China Beijing Inst Technol Sch Math & Stat Beijing 100081 Peoples R China Beijing Inst Technol Beijing Key Lab MCAACI Beijing 100081 Peoples R China

The octonion linear canonical transform (OCLCT) is a generalized form of the octonion Fourier transform (OFT), which in recent years has gradually become a new research area at the intersection of mathematics and signal processing. The study of these transforms not only enriches algebraic content but also provides tools for understanding geometric and physical phenomena in higher dimensions. In this work, we study the properties and potential applications of OCLCTs. First, we derive the differential properties and convolution theorem for the left-sided octonion linear canonical transform (LOCLCT). Second, by utilizing the properties and corresponding convolution theorem, we discuss and analyze 3-D linear time-invariant (LTI) systems. Finally, the examples and simulations provided in this study demonstrate the effectiveness of the proposed transform in capturing LOCLCT-frequency components, highlighting its enhanced flexibility and multiscale analysis capabilities.

关键词： Octonion linear canonical transform Differential property convolution theorem Multidimensional linear time-invariant systems

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