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Article · Mathematics · January 9, 2025

Quaternion Fractional Fourier Transform: Bridging Signal Processing and Probability Theory

Muhammad Adnan Samad, Yuanqing Xia, Saima Siddiqui, M. Younus Bhat, Didar Urynbassarova, Altyn Urynbassarova

Didar Urynbassarova
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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.

Mathematical Analysis and Transform MethodsAlgebraic and Geometric AnalysisMathematical functions and polynomialsfractional Fourier transformPlancherel formulaconjugate symmetry

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