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Uncertainty Principles for Wigner–Ville Distribution Associated with the Quaternion Offset Linear Canonical Transform

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Abstract

Wigner–Ville distribution (WVD) associated with the quaternion offset linear canonical transform (QOLCT) (WVD–QOLCT) is the known furthest generalization of the WVD in quaternion algebra. WVD–QOLCT is a hybrid transform that combines the flexibility and results of both WVD and QOLCT. Recently, some properties and classical Heisenberg's uncertainty principle (UP) have been derived for the two-dimensional (2D) two-sided WVD–QOLCT. This paper complements it by presenting conjugation symmetry and nonlinearity properties. Then, to characterize the simultaneous localization of a signal and its WVD–QOLCT, we establish different UPs for the 2D WVD–QOLCT, such as logarithmic UP, Hardy's UP, and Beurling's UP. In the end, by using the nonlinearity property, the applications of the 2D WVD–QOLCT in the linear frequency modulated signal detection are proposed.

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Acknowledgements

The authors thank the anonymous referees for their insightful remarks that helped to the improved version of this paper.

Funding

This work was partially supported by Beijing Municipal Natural Science Foundation under Grant L191004 and National Natural Science Foundation of China under Grant 62171025.

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Authors and Affiliations

  1. School of Information and Electronics, Beijing Institute of Technology, Beijing, China

    Didar Urynbassarova & Feng Zhang

  2. Beijing Key Laboratory of Fractional Signals and Systems, Beijing, China

    Didar Urynbassarova & Feng Zhang

  3. Ecole Normale Supérieure (ENS-Meknès), Moulay Ismail University, Toulal, Meknès, Morocco

    Youssef El Haoui

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  1. Didar Urynbassarova
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Correspondence to Didar Urynbassarova.

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Urynbassarova, D., El Haoui, Y. & Zhang, F. Uncertainty Principles for Wigner–Ville Distribution Associated with the Quaternion Offset Linear Canonical Transform. Circuits Syst Signal Process 42, 385–404 (2023). https://doi.org/10.1007/s00034-022-02127-y

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