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.

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

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 (CC BY-SA) license. This ensures that the content is freely available to all users and can be shared and adapted, provided appropriate credit is given and any adaptations are distributed under the same license.

Copyright Holder

The copyright of all 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). This applies to all published versions, including the HTML and PDF formats of the articles.

Author Rights

While the Departemen Matematika dan Sains Data FMIPA UNAND holds the copyright for all published content, authors retain important rights under the Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA). This license grants authors and users the following rights:

  • Reuse: Authors can reuse and distribute their work for any lawful purpose, including sharing on personal websites, institutional repositories, or in subsequent publications.
  • Attribution and Adaptation: Authors and others may remix, adapt, and build upon the published work for any purpose, even commercially, as long as proper credit is given to the original authors, and any derivative works are distributed under the same CC BY-SA license.

Creative Commons License (CC BY-SA)

Under the terms of the CC BY-SA 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, even commercially.

However, the following conditions apply:

  • Attribution: Users must give appropriate credit to the original author(s) and Departemen Matematika dan Sains Data FMIPA UNAND, provide a link to the license, and indicate if changes were made. Attribution must not imply endorsement by the author or the journal.
  • ShareAlike: If users remix, transform, or build upon the material, they must distribute their contributions under the same license as the original.

For more information about the CC BY-SA license, please visit the Creative Commons website.

Third-Party Content

If authors include third-party material (such as figures, tables, or images) that is not covered by a Creative Commons license, they must obtain the necessary permissions for reuse and provide proper attribution. Authors are required to ensure that any third-party content complies with open-access licensing requirements or includes permissions for redistribution under similar terms.

Copyright and Licensing Information Display

The copyright and licensing terms will be clearly displayed on each article's landing page, as well as within the full-text versions (HTML and PDF) of all published articles.

No "All Rights Reserved"

As an open-access journal, JMUA does not use "All Rights Reserved" policies. Instead, the CC BY-SA license ensures that the works remain accessible and reusable for a wide audience while still protecting both the authors' and the copyright holder's rights.

Â