ESSENTIAL PROPERTIES RELATED TO SHORT-TIME FRACTIONAL FOURIER TRANSFORM

Authors

  • SRI SULASTERI UIN Alauddin Makassar
  • NASRULLAH BACHTIAR
  • WAHYUNI EKASASMITA

DOI:

https://doi.org/10.25077/jmua.13.4.316-323.2024

Keywords:

fractional Fourier transform, short-time fractional Fourier transform, uncertainty principle

Abstract

We start by defining the short-time fractional Fourier transform in this paper, which is a natural generalization of the fractional Fourier transform. We then investigate its essential properties and explore an uncertainty principle related to this proposed transformation.

Downloads

Download data is not yet available.

References

Namias, V., 1980, The fractional order Fourier transform and its application to quantum mechanics,J. Inst. Math. Appl., Vol. 25: 241-265.

McBride, A.C., Kerr, F.H., 1987, On Namias’s fractional Fourier transforms, IMA J. Appl. Math., Vol 39: 159–175.

Tao, R., Xin, Y., and Wang, Y., 2007, Double image encryption based on random phase encoding in the fractional Fourier domain, Opt. Exp., Vol. 15: 16067–16079.

Tao, R., Lang, J., and Wang, Y., 2008, Optical image encryption based on the multiple-parameter fractional Fourier transform, Opt. Lett., Vol. 33: 581–583.

Qi, L., Tao, R., Zhou, S.Y., and Wang, Y., 2004, Detection and parameter es- timation of multicomponent LFM signal based on the fractional Fourier trans- form, Science in China (Ser. F. Information Sciences), Vol. 47: 184–198.

Pei, S.C., and Ding, J.J., 2000, Closed-form discrete fractional and affine Fourier transforms, IEEE Transactions on Signal Processing, Vol. 48(5): 1338–1353.

Durak, L., and Arikan, O., 2003, Short-time Fourier transform: Two fundamen- tal properties and an optimal implementation, IEEE Transactions on Signal Processing, Vol. 51(5): 1231–1242.

Bahri, M., 2017, Logarithmic uncertainty principle, convolution theorem re- lated to continuous fractional wavelet transform and its properties on a gener- alized Sobolev space, International Journal of Wavelets, Multiresolution and Information Processing, Vol. 15 (5): 1750050.

Anh, P.K., Castro, L.P., Thao, P.T., 2017, Two new convolutions for the frac- tional Fourier transform, Wireless Personal Communications, Vol. 92: 623-637.

Shi, J., Sha, X., Shong, X., and Zhang, N., 2012, Generalized convolution theorem associated with fractional Fourier transform, Wireless Communication and Mobile Computing, Vol. 14: 1340-1351.

Tao, R., Li, Y. L., Wang, Y., 2009, Short-time fractional Fourier transform and its applications, IEEE Transactions on Signal Processing, Vol. 58(5): 2568- 2580.

Tao, R., Qi L., and Wang, Y. 2004, Theory and Application of the Fractional Fourier Transform, Tsinghua University Press, Beijing.

Cohen, L., 2000, The uncertainty principles of windowed wave functions, Opt. Communication, 179(2000): 221–229.

Wo´dkiewicz, K., 1984, Operational approach to phase-space measurements in quantum mechanics, Phys. Rev. Lett., Vol. 52(13): 1064–1067.

Stankovic, L., Alieva, T., Bastiaans, M.J., 2003, Time–frequency signal analysis based on the windowed fractional Fourier transform, Signal Processing, Vol. 8: 2459–2468.

Guanlei, X., Xiaotong, W., Xiaogang, X., 2009, The logarithmic, Heisenberg’s and short-time uncertainty principles associated with fractional Fourier trans- form, Signal Processing, Vol. 89(3): 339-343.

Bahri, M., and Karim, S.A.A., 2023, Fractional Fourier transform: main pro- perties and inequalities, Mathematics, Vol. 11(5): 1234.

Downloads

Additional Files

Published

31-10-2024

How to Cite

SULASTERI, S., BACHTIAR, N., & EKASASMITA, W. (2024). ESSENTIAL PROPERTIES RELATED TO SHORT-TIME FRACTIONAL FOURIER TRANSFORM. Jurnal Matematika UNAND, 13(4), 316–323. https://doi.org/10.25077/jmua.13.4.316-323.2024

Issue

Vol. 13 No. 4 (2024)

Section

Articles

License

All articles published in Jurnal Matematika UNAND (JMUA) are open access and licensed under the Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA 4.0). This license allows published content to be freely accessed, shared, redistributed, and adapted for any purpose, including commercial use, provided that the applicable license conditions are followed.

Copyright Holder

The copyright of articles published in Jurnal Matematika UNAND is held by the Departemen Matematika dan Sains Data, Fakultas Matematika dan Ilmu Pengetahuan Alam (FMIPA), Universitas Andalas (UNAND), in accordance with the journal's stated copyright policy.

Author Rights

Under the CC BY-SA 4.0 license, authors may share, distribute, and reuse their published articles, subject to the applicable license conditions. These rights include:

  • Self-Archiving: Authors may deposit their published articles in institutional repositories, personal websites, or academic platforms.

  • Reuse: Authors may reuse their published work in subsequent publications, research, teaching materials, or other lawful purposes, provided that appropriate attribution is maintained.

  • Distribution: Authors may share and distribute their published articles without requiring additional permission under the terms of the license.

Creative Commons License (CC BY-SA 4.0)

Under the CC BY-SA 4.0 license, users are free to:

  • Share: Copy and redistribute the material in any medium or format.

  • Adapt: Remix, transform, and build upon the material for any purpose, including commercial purposes.

These permissions are subject to the following conditions:

  • Attribution: Users must give appropriate credit to the original author(s), retain relevant copyright notices, provide a link to the license, and indicate if changes were made. Attribution must not imply endorsement by the author(s), copyright holder, or journal.

  • ShareAlike: If users remix, transform, or build upon the material, they must distribute their contributions under the same CC BY-SA 4.0 license or a compatible license.

  • No Additional Restrictions: Users may not apply legal terms or technological measures that legally restrict others from exercising the rights granted by the license.

For more information about the CC BY-SA 4.0 license, please visit:

https://creativecommons.org/licenses/by-sa/4.0/

Third-Party Content

Third-party materials, including figures, tables, photographs, and illustrations, are covered by the article's CC BY-SA 4.0 license only when their inclusion under that license is authorized.

Materials excluded from the license must be clearly identified with appropriate copyright notices and attribution. Authors are responsible for obtaining any necessary permissions, while users must comply with the applicable licensing terms or permissions when reusing excluded materials.

Copyright and Licensing Information

Copyright and licensing information should be clearly displayed on each published article's landing page and included in the full-text versions (HTML and PDF), where applicable, to ensure transparency regarding copyright ownership and permitted reuse.