As information technology continues to advance rapidly, the field of signal processing is confronted with increasingly complex high-dimensional data processing. Traditional linear canonical transforms (LCT) show certain limitations when dealing with high-dimensional data. Therefore, researchers have begun to explore hypercomplex number fields, which can represent rotations in three-dimensional space more naturally and accurately, and have demonstrated significant advantages in signal processing tasks. Transformations based on quaternions and octonions, such as the quaternion Fourier transform (QFT) and the octonion Fourier transform (OFT), have been widely applied in areas such as signal detection, pattern recognition, and time-frequency analysis. Against this backdrop, this paper investigated the quaternion offset linear canonical transform (QOLCT) and its extension in the octonion domain—the offset octonion linear canonical transform (OOCLCT). First, the paper established the modulatory and differentiation properties of the QOLCT, which were missing in existing literature, laying a solid theoretical foundation for subsequent research. Then, based on the relationship between the QOLCT and the OOCLCT, multiple classical uncertainty principles within the framework of the OOCLCT were derived. These principles included the Heisenberg uncertainty principle, the sharp Hausdorff-Young inequality, the Matolcsi-Szucs uncertainty principle, and the Benedicks-Amrein-Berthier uncertainty principle. Through the derivation and verification of these uncertainty principles, this paper not only deepened the understanding of the OOCLCT theory but also provided new mathematical tools and methods for high-dimensional signal processing. Finally, we discussed the future research directions of the OOCLCT, providing a reference for subsequent studies.
Citation: Yixuan Lv, Qiang Feng, Bo Li, Jinyi Ji, Ziyuan Guo. The offset octonion linear canonical transform: Differential properties and uncertainty principle[J]. AIMS Mathematics, 2025, 10(12): 30905-30926. doi: 10.3934/math.20251356
As information technology continues to advance rapidly, the field of signal processing is confronted with increasingly complex high-dimensional data processing. Traditional linear canonical transforms (LCT) show certain limitations when dealing with high-dimensional data. Therefore, researchers have begun to explore hypercomplex number fields, which can represent rotations in three-dimensional space more naturally and accurately, and have demonstrated significant advantages in signal processing tasks. Transformations based on quaternions and octonions, such as the quaternion Fourier transform (QFT) and the octonion Fourier transform (OFT), have been widely applied in areas such as signal detection, pattern recognition, and time-frequency analysis. Against this backdrop, this paper investigated the quaternion offset linear canonical transform (QOLCT) and its extension in the octonion domain—the offset octonion linear canonical transform (OOCLCT). First, the paper established the modulatory and differentiation properties of the QOLCT, which were missing in existing literature, laying a solid theoretical foundation for subsequent research. Then, based on the relationship between the QOLCT and the OOCLCT, multiple classical uncertainty principles within the framework of the OOCLCT were derived. These principles included the Heisenberg uncertainty principle, the sharp Hausdorff-Young inequality, the Matolcsi-Szucs uncertainty principle, and the Benedicks-Amrein-Berthier uncertainty principle. Through the derivation and verification of these uncertainty principles, this paper not only deepened the understanding of the OOCLCT theory but also provided new mathematical tools and methods for high-dimensional signal processing. Finally, we discussed the future research directions of the OOCLCT, providing a reference for subsequent studies.
| [1] |
V. Namias, The fractional order Fourier transform and its application to quantum mechanics, J. Inst. Math. Appl., 25 (1980), 241–265. https://doi.org/10.1093/imamat/25.3.241 doi: 10.1093/imamat/25.3.241
|
| [2] |
I. M. Qasim, E. A. Mohammed, Optical image encryption based on linear canonical transform with sparse representation, Opt. Commun., 533 (2023), 129262. https://doi.org/10.1016/j.optcom.2023.129262 doi: 10.1016/j.optcom.2023.129262
|
| [3] |
B. Z. Li, R. Tao, Y. Wang, New sampling formulae related to linear canonical transform, Signal Process., 87 (2007), 983–990. https://doi.org/10.1016/j.sigpro.2006.09.008 doi: 10.1016/j.sigpro.2006.09.008
|
| [4] |
N. Le Bihan, S. J. Sangwine, T. A. Ell, Instantaneous frequency and amplitude of orthocomplex modulated signals based on quaternion Fourier transform, Signal Process., 94 (2014), 308–318. https://doi.org/10.1016/j.sigpro.2013.06.028 doi: 10.1016/j.sigpro.2013.06.028
|
| [5] |
G. L. Xu, X. T. Wang, X. G. Xu, Fractional quaternion Fourier transform, convolution and correlation, Signal Process., 88 (2008), 2511–2517. https://doi.org/10.1016/j.sigpro.2008.04.012 doi: 10.1016/j.sigpro.2008.04.012
|
| [6] |
B. J. Chen, M. Yu, Y. H. Tian, L. Li, D. C. Wang, X. M. Sun, Multiple-parameter fractional quaternion Fourier transform and its application in colour image encryption, IET Image Process., 12 (2018), 2238–2249. https://doi.org/10.1049/iet-ipr.2018.5440 doi: 10.1049/iet-ipr.2018.5440
|
| [7] |
K. I. Kou, J. Y. Ou, J. Morais, On uncertainty principle for quaternionic linear canonical transform, Abst. Appl. Anal., 1 (2013), 725952. https://doi.org/10.1155/2013/725952 doi: 10.1155/2013/725952
|
| [8] |
S. Saima, B. Z. Li, Quaternionic one-dimensional linear canonical transform, Optik, 244 (2021), 166914. https://doi.org/10.1016/j.ijleo.2021.166914 doi: 10.1016/j.ijleo.2021.166914
|
| [9] |
M. Y. Bhat, A. H. Dar, Quaternion offset linear canonical transform in one-dimensional setting, J. Anal., 31 (2023), 2613–2622. https://doi.org/10.1007/s41478-023-00585-4 doi: 10.1007/s41478-023-00585-4
|
| [10] |
D. Urynbassarova, A. A. Teali, Convolution, correlation, and uncertainty principles for the quaternion offset linear canonical transform, Mathematics, 11 (2023), 2201. https://doi.org/10.3390/math11092201 doi: 10.3390/math11092201
|
| [11] |
W. B. Gao, B. Z. Li, Quaternion windowed linear canonical transform of two-dimensional signals, Adv. Appl. Clifford Algebras, 30 (2020), 16. https://doi.org/10.1007/s00006-020-1042-4 doi: 10.1007/s00006-020-1042-4
|
| [12] |
H. H. Yang, Q. Feng, X. X. Wang, D. Urynbassarova, A. A. Teali, Reduced biquaternion windowed linear canonical transform: Properties and applications, Mathematics, 12 (2024), 743. https://doi.org/10.3390/math12050743 doi: 10.3390/math12050743
|
| [13] |
X. Yang, Q. Feng, N. Jiang, M. Y. Bhat, D. Urynbassarova, Properties and applications of octonion fractional Fourier transform for 3-D octonion signals, Digit. Signal Process., 165 (2025), 105339. https://doi.org/10.1016/j.dsp.2025.105339 doi: 10.1016/j.dsp.2025.105339
|
| [14] |
W. B. Gao, B. Z. Li, The octonion linear canonical transform: Definition and properties, Signal Process., 188 (2021), 108233. https://doi.org/10.1016/j.sigpro.2021.108233 doi: 10.1016/j.sigpro.2021.108233
|
| [15] |
N. Jiang, Q. Feng, X. Yang, J. R. He, B. Z. Li, The octonion linear canonical transform: Properties and applications, Chaos Soliton. Fract., 192 (2025), 116039. https://doi.org/10.1016/j.chaos.2025.116039 doi: 10.1016/j.chaos.2025.116039
|
| [16] |
Y. A. Bhat, N. A. Sheikh, Octonion offset linear canonical transform, Anal. Math. Phys., 12 (2022), 95. https://doi.org/10.1007/s13324-022-00705-6 doi: 10.1007/s13324-022-00705-6
|
| [17] |
G. B. Folland, A. Sitaram, The uncertainty principle: A mathematical survey, J. Fourier Anal. Appl., 3 (1997), 207–238. https://doi.org/10.1007/BF02649110 doi: 10.1007/BF02649110
|
| [18] |
A. H. Dar, M. Zayed, M. Y. Bhat, Short-time free metaplectic transform: Its relation to short-time Fourier transform in ${L^2}\left({{R^n}} \right)$ and uncertainty principles, AIMS Math., 8 (2023), 28951–28975. https://doi.org/10.3934/math.20231483 doi: 10.3934/math.20231483
|
| [19] |
P. Lian, Uncertainty principle for the quaternion Fourier transform, J. Math. Anal. Appl., 467 (2018), 1258–1269. https://doi.org/10.1016/j.jmaa.2018.08.002 doi: 10.1016/j.jmaa.2018.08.002
|
| [20] |
S. Shinde, V. M. Gadre, An uncertainty principle for real signals in the fractional Fourier transform domain, IEEE T. Signal Process., 49 (2001), 2545–2548. https://doi.org/10.1109/78.960402 doi: 10.1109/78.960402
|
| [21] |
Z. Aloui, K. Brahim, Fractional Fourier transform, signal processing and uncertainty principles, Circuits Syst. Signal Process., 42 (2023), 892–912. https://doi.org/10.1007/s00034-022-02138-9 doi: 10.1007/s00034-022-02138-9
|
| [22] |
F. Elgadiri, A. Akhlidj, Uncertainty principles for the fractional quaternion Fourier transform, J. Pseudo-Differ. Oper. Appl., 14 (2023), 54. https://doi.org/10.1007/s11868-023-00549-z doi: 10.1007/s11868-023-00549-z
|
| [23] |
F. Elgadiri, A. Akhlidj, Quantitative uncertainty principles associated with the fractional quaternion Fourier transform, J. Anal., 33 (2025), 2497–2519. https://doi.org/10.1007/s41478-025-00953-2 doi: 10.1007/s41478-025-00953-2
|
| [24] |
Q. Feng, B. Z. Li, J. M. Rassias Weighted Heisenberg-Pauli-Weyl uncertainty principles for the linear canonical transform, Signal Process., 165 (2019), 209–221. https://doi.org/10.1016/j.sigpro.2019.07.008 doi: 10.1016/j.sigpro.2019.07.008
|
| [25] |
W. B. Gao, New uncertainty principles in the linear canonical transform domains based on hypercomplex functions, Axioms, 14 (2025), 415. https://doi.org/10.3390/axioms14060415 doi: 10.3390/axioms14060415
|
| [26] |
M. Bahri, R. Ashino, Two-dimensional quaternion linear canonical transform: Properties, convolution, correlation, and uncertainty principle, J. Math., 1 (2019), 1062979. https://doi.org/10.1155/2019/1062979 doi: 10.1155/2019/1062979
|
| [27] |
X. Y. Zhu, S. Z. Zheng, On uncertainty principle for the two-sided quaternion linear canonical transform, J. Pseudo-Differ. Oper. Appl., 12 (2021), 3. https://doi.org/10.1007/s11868-021-00395-x doi: 10.1007/s11868-021-00395-x
|
| [28] |
M. Bahri, Generalized uncertainty principles for offset quaternion linear canonical transform, J. Anal., 32 (2024), 2525–2538. https://doi.org/10.1007/s41478-024-00733-4 doi: 10.1007/s41478-024-00733-4
|
| [29] |
M. Bahri, N. I. Tahir, N. Bachtiar, M. Zakir, Offset quaternion linear canonical transform: Properties, uncertainty inequalities and application, J. Franklin Inst., 362 (2025), 107553. https://doi.org/10.1016/j.jfranklin.2025.107553 doi: 10.1016/j.jfranklin.2025.107553
|
| [30] |
P. Lian, The octonionic Fourier transform: Uncertainty relations and convolution, Signal Process., 164 (2019), 295–300. https://doi.org/10.1016/j.sigpro.2019.06.015 doi: 10.1016/j.sigpro.2019.06.015
|
| [31] |
M. Zayed, Y. El Haoui, The uncertainty principle for the octonion Fourier transform, Math. Method. Appl. Sci., 46 (2023), 2651–2666. https://doi.org/10.1002/mma.8667 doi: 10.1002/mma.8667
|
| [32] | M. Y. Bhat, A. H. Dar, Uncertainty inequalities for 3D octonionic-valued signals associated with octonion offset linear canonical transform, 2021. https://doi.org/10.48550/arXiv.2111.11292 |
| [33] |
Z. C. Zhang, X. Y. Shi, A. Y. Wu, D. Li, Sharper N-D Heisenberg's uncertainty principle, IEEE Signal Process. Lett., 28 (2021), 1665–1669. https://doi.org/10.1109/LSP.2021.3101114 doi: 10.1109/LSP.2021.3101114
|
| [34] |
A. Iosevich, A. Mayeli, Uncertainty principles, restriction, Bourgain's ${\Lambda _q}$ theorem, and signal recovery, Appl. Comput. Harmon. Anal., 76 (2025), 101734. https://doi.org/10.1016/j.acha.2024.101734 doi: 10.1016/j.acha.2024.101734
|
Yixuan Lv, Qiang Feng, Bo Li, Jinyi Ji, Ziyuan Guo. The offset octonion linear canonical transform: Differential properties and uncertainty principle[J]. AIMS Mathematics, 2025, 10(12): 30905-30926. doi: 10.3934/math.20251356
DownLoad: