IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v13y2025i2p195-d1563225.html
   My bibliography  Save this article

Quaternion Fractional Fourier Transform: Bridging Signal Processing and Probability Theory

Author

Listed:
  • Muhammad Adnan Samad

    (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)

  • Yuanqing Xia

    (School of Automation, Beijing Institute of Technology, Beijing 100081, China
    Zhongyuan University of Technology, Zhengzhou 450007, China)

  • Saima Siddiqui

    (Department of Mathematics, Fergana Polytechnic Institute, Fergana 150100, Uzbekistan
    Computer Engineering and Artificial Intelligence Department, Tashkent University of Information Technologies, 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
    Institute of Automation and Information Technologies, Department of Cybersecurity, Information Processing and Storage, Satbayev University, Almaty 050000, Kazakhstan)

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.

Suggested Citation

  • Muhammad Adnan Samad & Yuanqing Xia & Saima Siddiqui & Muhammad Younus Bhat & Didar Urynbassarova & Altyn Urynbassarova, 2025. "Quaternion Fractional Fourier Transform: Bridging Signal Processing and Probability Theory," Mathematics, MDPI, vol. 13(2), pages 1-19, January.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:2:p:195-:d:1563225
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/13/2/195/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/13/2/195/
    Download Restriction: no
    ---><---

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:13:y:2025:i:2:p:195-:d:1563225. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.