Quaternion Fractional Fourier Transform: Bridging Signal Processing and Probability Theory

Muhammad Adnan Samad, Yuanqing Xia (), Saima Siddiqui, Muhammad Younus Bhat, Didar Urynbassarova and Altyn Urynbassarova
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Muhammad Adnan Samad: School of Automation, Beijing Institute of Technology, Beijing 100081, China
Yuanqing Xia: School of Automation, Beijing Institute of Technology, Beijing 100081, China
Saima Siddiqui: Department of Mathematics, Fergana Polytechnic Institute, Fergana 150100, Uzbekistan
Muhammad Younus Bhat: Department of Mathematical Sciences, Islamic University of Science and Technology, Kashmir, Awantipora 192122, India
Didar Urynbassarova: National Engineering Academy of the Republic of Kazakhstan, Almaty 050010, Kazakhstan
Altyn Urynbassarova: Faculty of Information Technology, Department of Information Security, L.N. Gumilyov Eurasian National University, Astana 010000, Kazakhstan

Mathematics, 2025, vol. 13, issue 2, 1-19

Abstract: 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.

Keywords: quaternion fractional Fourier transform; probability theory; quaternion algebra; characteristic function; stochastic processes; statistical analysis; quaternion-valued signals (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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