Short-time free metaplectic transform: Its relation to short-time Fourier transform in $ L^2(\mathbb R^n) $ and uncertainty principles

Aamir H. Dar; Mohra Zayed; M. Younus Bhat; Aamir H. Dar; Mohra Zayed; M. Younus Bhat

doi:10.3934/math.20231483

AIMS Mathematics

2023, Volume 8, Issue 12: 28951-28975. doi: 10.3934/math.20231483

Previous Article Next Article

Research article Special Issues

Short-time free metaplectic transform: Its relation to short-time Fourier transform in $ L^2(\mathbb R^n) $ and uncertainty principles

1.
Department of Mathematical Sciences, Islamic University of Science and Technology, Kashmir, 192122, India
2.
Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia

Received: 18 July 2023 Revised: 12 August 2023 Accepted: 18 August 2023 Published: 25 October 2023
MSC : 47B38, 42B10, 70H15, 65R10, 44A35

The free metaplectic transformation (FMT) has gained much popularity in recent times because of its various applications in signal processing, paraxial optical systems, digital algorithms, optical encryption and so on. However, the FMT is inadequate for localized analysis of non-transient signals, as such, it is imperative to introduce a unique localized transform coined as the short-time free metaplectic transform (ST-FMT). In this paper, we investigate the ST-FMT. First we propose the definition of the ST-FMT and provide the time-frequency analysis of the proposed transform in the FMT domain. Second we establish the relationship between the ST-FMT and short-time Fourier transform (STFT) in $ L^2(\mathbb R^n) $ and investigate the basic properties of the proposed transform including the reconstruction formula, Moyal's formula. The emergence of the ST-FMT definition and its properties broadens the development of time-frequency representation of higher-dimensional signals theory to a certain extent. We extend some different uncertainty principles (UPs) from quantum mechanics including Lieb's inequality, Pitt's inequality, Hausdorff-Young inequality, Heisenberg's UP, Hardy's UP, Beurling's UP, Logarithmic UP and Nazarov's UP. Finally, we give a numerical example and a possible applications of the proposed ST-FMT.
- short-time free metaplectic transform,
- Moyal's formula,
- uncertainty principle,
- Nazarov's UP,
- Hardy's UP,
- Logarithmic's UP
Citation: Aamir H. Dar, Mohra Zayed, M. Younus Bhat. Short-time free metaplectic transform: Its relation to short-time Fourier transform in $ L^2(\mathbb R^n) $ and uncertainty principles[J]. AIMS Mathematics, 2023, 8(12): 28951-28975. doi: 10.3934/math.20231483

Related Papers:

Abstract

The free metaplectic transformation (FMT) has gained much popularity in recent times because of its various applications in signal processing, paraxial optical systems, digital algorithms, optical encryption and so on. However, the FMT is inadequate for localized analysis of non-transient signals, as such, it is imperative to introduce a unique localized transform coined as the short-time free metaplectic transform (ST-FMT). In this paper, we investigate the ST-FMT. First we propose the definition of the ST-FMT and provide the time-frequency analysis of the proposed transform in the FMT domain. Second we establish the relationship between the ST-FMT and short-time Fourier transform (STFT) in $ L^2(\mathbb R^n) $ and investigate the basic properties of the proposed transform including the reconstruction formula, Moyal's formula. The emergence of the ST-FMT definition and its properties broadens the development of time-frequency representation of higher-dimensional signals theory to a certain extent. We extend some different uncertainty principles (UPs) from quantum mechanics including Lieb's inequality, Pitt's inequality, Hausdorff-Young inequality, Heisenberg's UP, Hardy's UP, Beurling's UP, Logarithmic UP and Nazarov's UP. Finally, we give a numerical example and a possible applications of the proposed ST-FMT.

References

[1]	G. B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, Princeton: Princeton University Press, 120 (1989). https://doi.org/10.1515/9781400882427
[2]	Z. C. Zhang, Unified Wigner-Ville distribution and ambiguity function in the linear canonical transform domain, Signal Process., 114 (2015), 45–60. https://doi.org/10.1016/j.sigpro.2015.02.016 doi: 10.1016/j.sigpro.2015.02.016
[3]	Z. C. Zhang, New Wigner distribution and ambiguity function based on the generalized translation in the linear canonical transform domain, Signal Process., 118 (2016), 51–61. https://doi.org/10.1016/j.sigpro.2015.06.010 doi: 10.1016/j.sigpro.2015.06.010
[4]	Z. C. Zhang, T. Yu, M. K. Luo, K. Deng, Estimating instantaneous frequency based on phase derivative and linear canonical transform with optimized computational speed, IET Signal Process., 12 (2018), 574–580. https://doi.org/10.1049/iet-spr.2017.0469 doi: 10.1049/iet-spr.2017.0469
[5]	Z. C. Zhang, J. Shi, X. P. Liu, L. He, M. Han, Q. Z. Li, et al., Sampling and reconstruction in arbitrary measurement and approximation spaces associated with linear canonical transform, IEEE Trans. Signal Process., 64 (2016), 6379–6391. https://doi.org/10.1109/TSP.2016.2602808 doi: 10.1109/TSP.2016.2602808
[6]	M. Moshinsky, C. Quesne, Linear canonical transformations and their unitary representations, J. Math. Phys., 12 (1971), 1772–1780. https://doi.org/10.1063/1.1665805 doi: 10.1063/1.1665805
[7]	F. A. Shah, A. Y. Tantary, Lattice-based multi-channel sampling theorem for linear canonical transform, Digit Signal Process., 117 (2021), 103168. https://doi.org/10.1016/j.dsp.2021.103168 doi: 10.1016/j.dsp.2021.103168
[8]	Z. Zhang, Uncertainty principle of complex-valued functions in specific free metaplectic transformation domains, J. Fourier Anal. Appl., 27 (2021). http://dx.doi.org/10.1007/s00041-021-09867-6
[9]	Z. Zhang, Uncertainty principle for real functions in free metaplectic transformation domains, J. Fourier Anal. Appl., 25 (2019), 2899–2922. https://doi.org/10.1007/s00041-019-09686-w doi: 10.1007/s00041-019-09686-w
[10]	M. Y. Bhat, A. H. Dar, Convolution and correlation theorems for Wigner-Ville distribution associated with the quaternion offset linear canonical transform, Signal Image Video P., 16 (2022), 1235–1242. https://doi.org/10.1007/s11760-021-02074-2 doi: 10.1007/s11760-021-02074-2
[11]	M. Y. Bhat, A. H. Dar, Wavelet packets associated with linear canonical transform on spectrum, Int. J. Wavelets Multi., 19 (2021). https://doi.org/10.1142/S0219691321500302
[12]	M. Y. Bhat, A. H. Dar, Fractional vector-valued nonuniform MRA and associated wavelet packets on $L^2(\mathbb R^2, C^M)$, Fract. Calc. Appl. Anal., 25 (2022), 687–719. https://doi.org/10.1007/s13540-022-00035-1 doi: 10.1007/s13540-022-00035-1
[13]	M. Y. Bhat, A. H. Dar, The algebra of 2D Gabor quaternionic offset linear canonical transform and uncertainty principles, J. Anal., 30 (2022), 637–649. https://doi.org/10.1007/s41478-021-00364-z doi: 10.1007/s41478-021-00364-z
[14]	M. Y. Bhat, A. H. Dar, Donoho starks and Hardys uncertainty principles for the short-time quaternion offset linear canonical transform, Filomat, 37 (2023), 4467– 4480.
[15]	W. Heisenberg, Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift Physik, 43 (1927), 172–198. https://doi.org/10.1007/BF01397280 doi: 10.1007/BF01397280
[16]	D. L. Donoho, P. B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math., 49 (1989), 906–931. https://doi.org/10.1137/0149053 doi: 10.1137/0149053
[17]	B. Ricaud, B. A. Torrésani, Survey of uncertainty principles and some signal processing applications, Adv. Comput. Math., 40 (2014), 629–650. https://doi.org/10.1007/s10444-013-9323-2 doi: 10.1007/s10444-013-9323-2
[18]	Q. Zhang, Zak transform and uncertainty principles associated with the linear canonical transform, IET Signal Process., 10 (2016), 791–797. https://doi.org/10.1049/iet-spr.2015.0514 doi: 10.1049/iet-spr.2015.0514
[19]	H. Y. Huo, Uncertainty principles for the offset linear canonical transform, Circ. Syst. Signal Process., 38 (2019), 395–406. https://doi.org/10.1007/s00034-018-0863-z doi: 10.1007/s00034-018-0863-z
[20]	Z. C. Zhang, N-dimensional Heisenberg's uncertainty principle for fractional Fourier transform, Phys. Eng. Math., 2019. Available from: https://arXiv.org/abs/1906.05451.
[21]	Y. G. Li, B. Z. Li, H. F. Sun, Uncertainty principles for Wigner-Ville distribution associated with the linear canonical transforms, Abstr. Appl. Anal., 2014 (2014), 1–9. https://doi.org/10.1155/2014/470459 doi: 10.1155/2014/470459
[22]	F. L. Nazarov, Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type, Algebra I Anal. 5 (1993), 663–717.
[23]	P. Jaming, Nazarov's uncertainty principle in highter dimension, J. Approx. Theory, 149 (2007), 30–41. https://doi.org/10.1016/j.jat.2007.04.005 doi: 10.1016/j.jat.2007.04.005
[24]	A. Bonami, B. Demange, P. Jaming, Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms, Rev. Mat. Iberoam., 19 (2003), 23–55. https://doi.org/10.4171/RMI/337 doi: 10.4171/RMI/337
[25]	S. C. Bagchi, S. K. Ray, Uncertainty principles like Hardy's theorem on some Lie groups, J. Aust. Math. Soc., 1999,289–302. https://doi.org/10.1017/S1446788700035886
[26]	W. Beckner, Pitt's inequality and the uncertainty principle, Proc. Am. Math. Soc., 123 (1995), 1897–1905. https://doi.org/10.1090/S0002-9939-1995-1254832-9 doi: 10.1090/S0002-9939-1995-1254832-9
[27]	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
[28]	M. Y. Bhat, A. H. Dar, Uncertainty principles for quaternion linear canonical S-transform, Funct. Anal., 2021.
[29]	M. Y. Bhat, A. H. Dar, Uncertainty inequalities for 3D octonionic-valued signals associated with octonion offset linear canonical transform, Signal Process., 2021.
[30]	M. Y. Bhat, Windowed linear canonical transform: Its relation to windowed Fourier transform and uncertainty principles, J. Inequal. Appl., 4 (2022). https://doi.org/10.1186/s13660-021-02737-1
[31]	M. Y. Bhat, A. H. Dar, Octonion spectrum of 3D short-time LCT signals, Signal Process., 2021. https://doi.org/10.1016/j.ijleo.2022.169156
[32]	R. Jing, B. Liu, R. Li, R. Liu, The N-dimensional uncertainty principle for the free metaplectic transformation, Mathematics, 8 (2020). http://dx.doi.org/10.3390/math8101685
[33]	H. M. Srivastava, F. A. Shah, W. Z. Lone, Non-separable linear canonical wavelet transform, Symmetry, 13 (2021), 2182. https://doi.org/10.3390/ sym13112182 doi: 10.3390/sym13112182
[34]	M. Bahri, R. Ashino, Some properties of windowed linear canonical transform and its logarithmic uncertainty principle, Int. J. Wavelets Multi., 14 (2016), 1650015. https://doi.org/10.1142/S0219691316500156 doi: 10.1142/S0219691316500156
[35]	F. A. Shah, A. Y. Tantary, Multi-dimensional linear canonical transform with applications to sampling andmultiplicative filtering, Multidim. Syst. Sign. P., 117 (2021) 103168. https://doi.org/10.1007/s11045-021-00816-6
[36]	M. Bahri, Windowed linear canonical transform: Its relation to windowed Fourier transform and uncertainty principles, J. Inequal. Appl., 2022 (2022). https://doi.org/10.1186/s13660-021-02737-1
[37]	G. H. Hardy, A theorem concerning Fourier transforms, J. Lond. Math. Soc., 8 (1933), 227–231. https://doi.org/10.1112/jlms/s1-8.3.227 doi: 10.1112/jlms/s1-8.3.227
[38]	L. Escauriaza, C. E. Kenig, G. Ponce, L. Vega, The sharp Hardy uncertainty principle for Schodinger evolutions, Duke Math. J., 155 (2010), 163–187. https://doi.org/10.1215/00127094-2010-053 doi: 10.1215/00127094-2010-053
[39]	Y. J. Cao, B. Z. Li, Y. G. Li, Y. H. Chen, Logarithmic uncertainty relations for odd or even signals associate with Wigner-Ville distribution, Circ. Syst. Signal Pr., 35 (2016), 2471–2486. https://doi.org/10.1007/s00034-015-0146-x doi: 10.1007/s00034-015-0146-x
[40]	R. Skoog, An uncertainty principle for functions vanishing on a half-line, IEEE T. Circ. Theory, 173 (1970), 241–243. https://doi.org/10.1109/TCT.1970.1083079 doi: 10.1109/TCT.1970.1083079

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)