Abstract
The main purpose of this paper is to introduce a family of convolution-based generalized Stockwell transforms in the context of time-fractional-frequency analysis. The spirit of this article is completely different from two existing studies (see D. P. Xu and K. Guo [Appl. Geophys. 9 (2012) 73–79] and S. K. Singh [J. Pseudo-Differ. Oper. Appl. 4 (2013) 251–265]) in the sense that our approach completely relies on the convolution structure associated with the fractional Fourier transform. We first study all of the fundamental properties of the generalized Stockwell transform, including a relationship between the fractional Wigner distribution and the proposed transform. In the sequel, we introduce both the semi-discrete and discrete counterparts of the proposed transform. We culminate our investigation by establishing some Heisenberg-type inequalities for the generalized Stockwell transform in the fractional Fourier domain.
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References
Gabor, D.: Theory of communications. J. Inst. Electr. Eng. 93, 429–457 (1946)
Debnath, L., Shah, F.A.: Wavelet Transforms and Their Applications. Birkhäuser, Boston (2015)
Debnath, L., Shah, F.A.: Lectuer Notes on Wavelet Transforms. Birkhäuser, Boston (2017)
Duabechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)
Srivastava, H.M., Khatterwani, K., Upadhyay, S.K.: A certain family of fractional wavelet transformations. Math. Methods Appl. Sci. 42, 3103–3122 (2019)
Stockwell, R.G., Mansinha, L., Lowe, R.P.: Localization of the complex spectrum: 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)
Ma, J., Jiang, J.: Analysis and design of modified window shapes for S-transform to improve time-frequency localization. Mech. Syst. Signal Process. 58, 271–284 (2015)
Xu, D.P., Guo, K.: Fractional \(S\)-transform-part 1: theory. Appl. Geophys. 9, 73–79 (2012)
Du, Z.C., Xu, D.P., Zhang, J.M.: Fractional \(S\)-transform-part 2: application to reservoir prediction and fluid identification. Appl. Geophys. 13, 343–352 (2016)
Akila, L., Roopkumar, R.: Quaternionic stockwell transform. Integr. Trans. Spec. Funct. 27, 484–504 (2016)
Hutniková, M., Mis̆ková, A.: Continuous Stockwell transform: Coherent states and localization operators, J. Math. Phys. 56, 1–15 (2015)
Riba, L., Wong, M.: Continuous inversion formulas for multi-dimensional modified Stockwell transforms. Integr. Trans. Spec. Funct. 26, 9–19 (2015)
Battisti, U., Riba, L.: Window-dependent bases for efficient representations of the Stockwell transform. Appl. Comput. Harmon. Anal. 40, 292–320 (2016)
Drabycz, S., Stockwell, R.G., Mitchell, J.R.: Image texture characterization using the discrete orthonormal \(S\)-transform. J. Digit. Imaging. 22, 696–708 (2009)
Moukadem, A., Bouguila, Z., Abdeslam, D.O., Dieterlen, A.: A new optimized Stockwell transform applied on synthetic and real non-stationary signals. Digit. Signal Process. 46, 226–238 (2015)
Shah, F.A., Tantary, A.Y.: Non-isotropic angular Stockwell transform and the associated uncertainty principles. Appl. Anal. (2019). https://doi.org/10.1080/00036811.2019.1622681
Shah, F.A., Tantary, A.Y.: Linear canonical Stockwell transform. J. Math. Anal. Appl. 484, 123673 (2020)
Dai, H.Z., Zheng, Z.B., Wang, W.: A new fractional wavelet transform. Commun. Nonlinear Sci. Numer. Simulat. 44, 19–36 (2017)
Singh, S.K.: Fractional \(S\)-transform on spaces of type \(W\). J. Pseudo-Differ. Oper. Appl. 4, 251–265 (2013)
Almeida, L.B.: The fractional Fourier transform and time-frequency representations. IEEE Trans. Signal Process. 42, 3084–3091 (1994)
Zhang, Z.C.: Novel Wigner distribution and ambiguity function associated with the linear canonical transform. Optik 127, 4995–5012 (2016)
Folland, G.B., Sitaram, A.: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3, 207–238 (1997)
Beckner, W.: Pitt’s inequality and the uncertainty principle. Proc. Am. Math. Soc. 123, 1897–1905 (1995)
Battle, G.: Heisenberg inequalities for wavelet states. Appl. Comput. Harmon. Anal. 4, 119–146 (1997)
Wilczok, E.: New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform. Doc. Math. 5, 201–226 (2000)
Xu, G.L., Wang, X.T., Xu, X.G.: The logarithmic, Heisenberg’s and short-time uncertainty principles associated with fractional Fourier transform. Signal Process. 89, 339–343 (2009)
Su, Y.: Heisenberg type uncertainty principle for continuous Shearlet transform. J. Nonlinear Sci. Appl. 9, 778–786 (2016)
Bahri, M., Shah, F.A., Tantary, A.Y.: Uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions. Integ. Transf. Special Funct. (2020). https://doi.org/10.1080/10652469.2019.1707816
Shah, F.A., Tantary, A.Y.: Polar wavelet transform and the associated uncertainty principles. Int. J. Theor. Phys. 57, 1774–1786 (2018)
Acknowledgements
The authors would like to thank the esteemed editor and the referee for their valuable comments and suggestions. The second-named author was financially supported by the Science and Engineering Research Board, Department of Science and Technology, Government of India under Grant No. EMR/2016/007951.
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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). https://doi.org/10.1007/s11868-020-00363-x
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DOI: https://doi.org/10.1007/s11868-020-00363-x
Keywords
- Stockwell transform
- Wavelet transform
- Wigner distribution
- Fractional Fourier transform
- Time-fractional-frequency analysis
- Uncertainty principle