Abstract
In the framework of higher-dimensional time-frequency analysis, we propose a novel Clifford-valued Stockwell transform for an effective and directional representation of Clifford-valued functions. The proposed transform rectifies the windowed Fourier and wavelet transformations by employing an angular, scalable and localized window, which offers directional flexibility in the multi-scale signal analysis in Clifford domains. The basic properties of the proposed transform such as inner product relation, reconstruction formula, and the range theorem are investigated using the machinery of operator theory and Clifford Fourier transforms. Moreover, several extensions of the well-known Heisenberg-type inequalities are derived for the proposed transform in the Clifford Fourier domain. We culminate our investigation by deriving the directional uncertainty principles for the Clifford-valued Stockwell transform. To validate the acquired results, illustrative examples are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bahri, M., Hitzer, E.: Clifford Fourier transform on multivector fields and uncertainty principles for dimensions \(n = 2({{\rm mod}}\, 4)\) and \(n = 3({{\rm mod}}\, 4)\). Adv. Appl. Clifford Algebra 18, 715–736 (2008)
Bahri, M., Adji, S., Zhao, J.: Clifford algebra-valued wavelet transform on multivector fields. Adv. Appl. Clifford Algebras 21(1), 13–30 (2011)
Bahri, M.: Clifford windowed Fourier transform applied to linear time-varying systems. Appl. Math. Sci. 6(58), 2857–2864 (2012)
Beckner, W.: Pitt’s inequality and the uncertainty principle. Proc. Am. Math. Soc. 123, 1897–1905 (1995)
Brackx, F., De Schepper, N., Sommen, F.: The Clifford Fourier transform. J. Fourier Anal. Appl. 6(11), 668–681 (2005)
Brackx, F., Hitzer, E., Sangwine, S.: History of quaternion and Clifford-Fourier transforms and wavelets. In: Hitzer, E., Sangwine, S. (eds.) Quaternion and Clifford Fourier Transforms and Wavelets. Trends in Mathematics, vol. 27, pp. XI–XXVII. Birkhäuser, Basel (2013)
Daubechies, I.: Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia (1992)
Debnath, L., Shah, F.A.: Wavelet Transforms and Their Applications. Birkhäuser, Boston (2015)
Folland, G.B., Sitaram, A.: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3, 207–238 (1997)
Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)
Guo, Q., Molahajloo, S., Wong, M.W.: Modified Stockwell transforms and time-frequency analysis. In: Rodino, L., Wong, M.W. (eds.) New Developments in Pseudo-Differential Operators. Operator Theory: Advances and Applications, vol. 189, pp. 275–285. Birkhäuser, Basel (2009)
Hitzer, E., Nitta, T., Kuroe, Y.: Applications of Clifford’s geometric algebra. Adv. Appl. Clifford Algebras 23, 377–404 (2013). https://doi.org/10.1007/s00006-013-0378-4
Hutniková, M., Misková, A.: Continuous Stockwell transform: coherent states and localization operators. J. Math. Phys. 56, 073504 (2015)
Lian, P.: Sharp inequalities for geometric Fourier transform and associated ambiguity function. J. Math. Anal. Appl. (2019). https://doi.org/10.1016/j.jmaa.2019.123730
Liu, Y., Wong, M.: Inversion formulas for two dimensional Stockwell transforms. In: Rodino, L., Schulze, B.W., Wong, M.W. (eds.) Pseudo Differential Operators: Partial Differential Equations and Time-Frequency Analysis, Fields Institute Communications, vol. 52, pp. 323–330. American Mathematical Society, Providence (2007)
Riba, L., Wong, M.W.: Continuous inversion formulas for multi-dimensional Stockwell transforms. Math. Model. Nat. Phenom. 8, 215–229 (2013)
Riba, L., Wong, M.: Continuous inversion formulas for multi-dimensional modified Stockwell transforms. Integral Transforms Spec. Funct. 26(1), 9–19 (2015)
Shah, F.A., Tantary, A.Y.: Non-isotropic angular Stockwell transform and the associated uncertainty principles. Appl. Anal. 100(4), 835–859 (2021)
Shah, F.A., Tantary, A.Y.: Linear canonical Stockwell transform. J. Math. Anal. Appl. 484(1) (2020). https://doi.org/10.1016/j.jmaa.2019.123673
Shah, F.A., Teali, A.A., Tantary, A.Y.: Linear canonical wavelet transform in quaternion domains. Adv. Appl. Clifford Algebras 31, 42 (2021)
Sommer, G.: Geometric Computing with Clifford Algebras. Springer, Berlin (2001)
Srivastava, H.M., Shah, F.A., Tantary, A.Y.: A family of convolution-based generalized Stockwell transforms. J. Pseudo Differ. Oper. Appl. 11, 1505–1536 (2020)
Stockwell, R.G., Mansinha, L., Lowe, R.P.: Localization of the complex sectrum: the \(S\) transform. IEEE Trans. Signal Process. 44, 998–1001 (1996)
Stockwell, R.G.: A basis for efficient representation of the \(S\)-transform. Digit. Signal Process. 17, 371–393 (2007)
Acknowledgements
F. A. Shah is supported by SERB (DST), Government of India under Grant no. EMR/2016/007951.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no competing interests.
Additional information
Communicated by Rafal Ablamowicz.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Shah, F.A., Teali, A.A. & Bahri, M. Clifford-Valued Stockwell Transform and the Associated Uncertainty Principles. Adv. Appl. Clifford Algebras 32, 25 (2022). https://doi.org/10.1007/s00006-022-01204-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00006-022-01204-4