Quaternion Fractional Fourier Transform: Bridging Signal Processing and Probability Theory
Samad M.A. Xia Y. Siddiqui S. Bhat M.Y. Urynbassarova D. Urynbassarova A.
January 2025Multidisciplinary Digital Publishing Institute (MDPI)
Mathematics
2025#13Issue 2
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.
characteristic function , probability theory , quaternion algebra , quaternion fractional Fourier transform , quaternion-valued signals , statistical analysis , stochastic processes
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School of Automation, Beijing Institute of Technology, Beijing, 100081, China
Electrical Engineering, Electrical Mechanics and Electrical Technologies Department, Fergana Polytechnic Institute, Fergana, 150100, Uzbekistan
Telecommunication Engineering Department, Tashkent University of Information Technologies, Fergana, 150100, Uzbekistan
Zhongyuan University of Technology, Zhengzhou, 450007, China
Department of Mathematics, Fergana Polytechnic Institute, Fergana, 150100, Uzbekistan
Computer Engineering and Artificial Intelligence Department, Tashkent University of Information Technologies, Fergana, 150100, Uzbekistan
Department of Mathematical Sciences, Islamic University of Science and Technology, Awantipora, Kashmir, 192122, India
National Engineering Academy of the Republic of Kazakhstan, Almaty, 050010, Kazakhstan
Faculty of Information Technology, Department of Information Security, L.N. Gumilyov, Eurasian National University, Astana, 010000, Kazakhstan
Institute of Automation and Information Technologies, Department of Cybersecurity, Information Processing and Storage, Satbayev University, Almaty, 050000, Kazakhstan
School of Automation
Electrical Engineering
Telecommunication Engineering Department
Zhongyuan University of Technology
Department of Mathematics
Computer Engineering and Artificial Intelligence Department
Department of Mathematical Sciences
National Engineering Academy of the Republic of Kazakhstan
Faculty of Information Technology
Institute of Automation and Information Technologies
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