Quaternion Fractional Fourier Transform: Bridging Signal Processing and Probability Theory
The one-dimensional quaternion fractional Fourier transform (1DQFRFT) introduces a fractional-order parameter that extends traditional Fourier transform techniques, providing new insights into the analysis of quaternion-valued signals. This paper presents a rigorous theoretical foundation for the 1D...
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| Format: | Article |
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2025-01-01
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| Series: | Mathematics |
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| author | Muhammad Adnan Samad Yuanqing Xia Saima Siddiqui Muhammad Younus Bhat Didar Urynbassarova Altyn Urynbassarova |
| author_facet | Muhammad Adnan Samad Yuanqing Xia Saima Siddiqui Muhammad Younus Bhat Didar Urynbassarova Altyn Urynbassarova |
| author_sort | Muhammad Adnan Samad |
| collection | DOAJ |
| description | The one-dimensional quaternion fractional Fourier transform (1DQFRFT) introduces a fractional-order parameter that extends traditional Fourier transform techniques, providing new insights into the analysis of quaternion-valued signals. This paper presents a rigorous theoretical foundation for the 1DQFRFT, examining essential properties such as linearity, the Plancherel theorem, conjugate symmetry, convolution, and a generalized Parseval’s theorem that collectively demonstrate the transform’s analytical power. We further explore the 1DQFRFT’s unique applications to probabilistic methods, particularly for modeling and analyzing stochastic processes within a quaternionic framework. By bridging quaternionic theory with probability, our study opens avenues for advanced applications in signal processing, communications, and applied mathematics, potentially driving significant advancements in these fields. |
| format | Article |
| id | doaj-art-103e97f0b56f47a8945a67cec58ea9c3 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-103e97f0b56f47a8945a67cec58ea9c32025-01-24T13:39:42ZengMDPI AGMathematics2227-73902025-01-0113219510.3390/math13020195Quaternion Fractional Fourier Transform: Bridging Signal Processing and Probability TheoryMuhammad Adnan Samad0Yuanqing Xia1Saima Siddiqui2Muhammad Younus Bhat3Didar Urynbassarova4Altyn Urynbassarova5School of Automation, Beijing Institute of Technology, Beijing 100081, ChinaSchool of Automation, Beijing Institute of Technology, Beijing 100081, ChinaDepartment of Mathematics, Fergana Polytechnic Institute, Fergana 150100, UzbekistanDepartment of Mathematical Sciences, Islamic University of Science and Technology, Kashmir, Awantipora 192122, IndiaNational Engineering Academy of the Republic of Kazakhstan, Almaty 050010, KazakhstanFaculty of Information Technology, Department of Information Security, L.N. Gumilyov Eurasian National University, Astana 010000, KazakhstanThe one-dimensional quaternion fractional Fourier transform (1DQFRFT) introduces a fractional-order parameter that extends traditional Fourier transform techniques, providing new insights into the analysis of quaternion-valued signals. This paper presents a rigorous theoretical foundation for the 1DQFRFT, examining essential properties such as linearity, the Plancherel theorem, conjugate symmetry, convolution, and a generalized Parseval’s theorem that collectively demonstrate the transform’s analytical power. We further explore the 1DQFRFT’s unique applications to probabilistic methods, particularly for modeling and analyzing stochastic processes within a quaternionic framework. By bridging quaternionic theory with probability, our study opens avenues for advanced applications in signal processing, communications, and applied mathematics, potentially driving significant advancements in these fields.https://www.mdpi.com/2227-7390/13/2/195quaternion fractional Fourier transformprobability theoryquaternion algebracharacteristic functionstochastic processesstatistical analysis |
| spellingShingle | Muhammad Adnan Samad Yuanqing Xia Saima Siddiqui Muhammad Younus Bhat Didar Urynbassarova Altyn Urynbassarova Quaternion Fractional Fourier Transform: Bridging Signal Processing and Probability Theory Mathematics quaternion fractional Fourier transform probability theory quaternion algebra characteristic function stochastic processes statistical analysis |
| title | Quaternion Fractional Fourier Transform: Bridging Signal Processing and Probability Theory |
| title_full | Quaternion Fractional Fourier Transform: Bridging Signal Processing and Probability Theory |
| title_fullStr | Quaternion Fractional Fourier Transform: Bridging Signal Processing and Probability Theory |
| title_full_unstemmed | Quaternion Fractional Fourier Transform: Bridging Signal Processing and Probability Theory |
| title_short | Quaternion Fractional Fourier Transform: Bridging Signal Processing and Probability Theory |
| title_sort | quaternion fractional fourier transform bridging signal processing and probability theory |
| topic | quaternion fractional Fourier transform probability theory quaternion algebra characteristic function stochastic processes statistical analysis |
| url | https://www.mdpi.com/2227-7390/13/2/195 |
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