The fractional fourier transform
Open accessAbstract
A brief introduction to the fractional Fourier transform and its properties is given. Its relation to phase-space representations (time- or space-frequency representations) and the concept of fractional Fourier domains are discussed. An overview of applications which have so far received interest are given and some potential application areas remaining to be explored are noted.
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The burst of publications on the fractional Fourier transform since the 1990s was mainly due to its optical interpretation and applications in optics, culminating in the book of Ozaktas et al.
“but it was mainly the optical interpretation and the applications in optics that gave a burst of publications since the 1990's that culminated in the book of Ozaktas et al [17].”
From Computation of the fractional Fourier transform (2004) · cites this paper as [17]
The introductory history passage (section “+”, page unavailable) says interest grew after researchers in optics and signal processing reintroduced the transform and recognized applications in a variety of areas. The Applications section reports substantial interest in optics, and References lists the 2001 Ozaktas et al. book. The paper does not say optics was the main cause of the publication burst or that the burst culminated in that book. (LLM, conf=0.85)
Experimental — accuracy not yet measuredThe fractional Fourier transform was later used in quantum mechanics and signal processing.
“Later on it was used in quantum mechanics [14,13] and signal processing [1], but it was mainly the optical interpretation and the applications in optics that gave a burst of publications since the 1990's that culminated in the book of Ozaktas et al [17].”
From Computation of the fractional Fourier transform (2004) · cites this paper as [17]
When ordered appropriately, the interlacing in step 6 of the algorithm corresponds to ordering the eigenvectors according to their number of zero crossings, which implies that the evens and odds will interlace.
“A careful mathematical analysis for which we refer to the literature [7,8,18,17] reveals that when ordered appropriately, the interlacing in step 6 of our algorithm in section 6 corresponds to ordering the eigenvectors according to their number of zero crossings. That implies for example that the evens and odds will interlace.”
From Computation of the fractional Fourier transform (2004) · cites this paper as [17]
The paper’s “2 Definition” section discusses Fourier-transform eigenfunctions, but the passages found do not address ordering eigenvectors by zero crossings or interlacing even and odd eigenvectors. Searches for those terms and for the algorithm’s step 6 found no passage establishing the attributed claim; page numbers are unavailable in our copy. (LLM, conf=0.72)
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Cites
The discrete fractional Fourier transform
Candan, C., Kutay, M. A., & Ozaktas, H. M. (2000) · IEEE Transactions on Signal Processing · doi:10.1109/78.839980 · Cited by 683 · see citing works
Self-citation
Flagged first, then by citations
Research works printed but not linked: 12
- H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay. The Frac- tional Fourier Transform with Applications in Optics and Signal Processing. John Wiley & Sons, 2001.
- H. M. Ozaktas, M. A. Kutay, and D. Mendlovic. Intro- duction to the fractional Fourier transform and its applica- tions. In Advances in Imaging and Electron Physics 106, Academic Press, San Diego, California, 1999. Pages 239- 291.
- D. A. Mustard. The fractional Fourier transform and a new uncertainty principle. School of Mathematics Preprint AM87/14, The University of New South Wales, Kensington, Australia, 1987.
- L. B. Almeida. The fractional Fourier transform and time- frequency representations. IEEE Trans Signal Processing, 42:3084-3091, 1994.
- F. Hlawatsch and G. F. Boudreaux-Bartels. Linear and quadratic time-frequency signal representations. IEEE Signal Processing Magazine, April 1992, pages 21-67.
- A. W. Lohmann, D. Mendlovic, and Z. Zalevsky. Frac- tional transformations in optics. In Progress in Op- tics XXXVIII, Elsevier, Amsterdam, 1998. Chapter IV, pages 263-342.
- M. A. Kutay, H. M. Ozaktas, O. Arıkan, and L. Onu- ral. Optimal filtering in fractional Fourier domains. IEEE Trans Signal Processing, 45:1129-1143, 1997.
- D. Mendlovic, Z. Zalevsky, and H. M. Ozaktas. Applica- tions of the fractional Fourier transform to optical pattern recognition. In Optical Pattern Recognition, Cambridge University Press, Cambridge, 1998. Chapter 4, pages 89- 125.
- O. Akay and G. F. Boudreaux-Bartels. Broadband inter- ference excision in spread spectrum communication sys- tems via fractional Fourier transform. In Proc 32nd Asilo- mar Conf Signals, Systems, Computers, IEEE, Piscat- away, New Jersey, 1998. Pages 832-837.
- M. J. Bastiaans and A. J. van Leest. From the rectangu- lar to the quincunx Gabor lattice via fractional Fourier transformation. IEEE Signal Processing Lett, 5:203-205, 1998.
- G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti. A unified framework for the fractional Fourier transform. IEEE Trans Signal Processing, 46:3206-3219, 1998.
- H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagı. Digital computation of the fractional Fourier transform. IEEE Trans Signal Processing, 44:2141-2150, 1996.
Cited by
Sorted by citations
Short-time fourier transform: two fundamental properties and an optimal implementation
Durak, L., & Arikan, O. (2003) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2003.810293 · Cited by 345 · see citing works
Computation of the fractional Fourier transform
Bultheel, A., & Martı́nez Sulbaran, H. E. (2004) · Applied and Computational Harmonic Analysis · doi:10.1016/j.acha.2004.02.001 · Cited by 150 · see citing works
Prajna, K., & Mukhopadhyay, C. K. (2020) · Journal of Nondestructive Evaluation · doi:10.1007/s10921-020-0658-6 · Cited by 38 · see citing works
Vilardy, J. M., Millán, M. S., & Pérez–Cabré, E. (2019) · Journal of Physics: Conference Series · doi:10.1088/1742-6596/1221/1/012046 · Cited by 5 · see citing works
- Authors
- Ozaktas, H. M., & Kutay, M. A.
- Affiliations
- Department of Electrical Engineering, Bilkent University, 06533 Bilkent, Ankara, Turkey
- Türkiye Bilimsel ve Teknolojik Araştırma Kurumu (TR)
- ORCID
- Kutay, M. Alper
- Venue
- 2001 European Control Conference (ECC)
- Publisher
- IEEE
- Published
- 2001-09
- Pages
- 1477-1483
- OpenAlex
- W1568450818
- Cited by
- 580
- References
- 1
- Citations by year
- 2026: 25 · 2025: 40 · 2024: 30 · 2023: 26 · 2022: 23 · 2021: 20 · 2020: 27 · 2019: 25 · 2018: 26 · 2017: 24 · 2016: 29 · 2015: 23 · 2014: 34 · 2013: 21 · 2012: 26
- FWCI
- 38.27
- Type
- conference paper
- Language
- en
- Open access
- green
- hdl.handle.net
- Locations
- European Control Conference (ECC) (publishedVersion)
- Bilkent University Institutional Repository (Bilkent University) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical Analysis and Transform Methods
- Physical Sciences › Mathematics › Modeling and Simulation › Fractional Differential Equations Solutions
- Physical Sciences › Computer Science › Signal Processing › Digital Filter Design and Implementation
- Keywords
- fractional Fourier transform, phase-space representation, time-frequency representation, signal processing applications
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