Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms
Open accessAbstract
A concise introduction to the concept of fractional Fourier transforms is followed by a discussion of their relation to chirp and wavelet transforms. The notion of fractional Fourier domains is developed in conjunction with the Wigner distribution of a signal. Convolution, filtering, and multiplexing of signals in fractional domains are discussed, revealing that under certain conditions one can improve on the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing.
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Cites
The Fractional Order Fourier Transform and its Application to Quantum Mechanics
NAMIAS, V. (1980) · IMA Journal of Applied Mathematics · doi:10.1093/imamat/25.3.241 · Cited by 1,722 · see citing works
Image rotation, Wigner rotation, and the fractional Fourier transform
Lohmann, A. W. (1993) · Journal of the Optical Society of America A · doi:10.1364/josaa.10.002181 · Cited by 1,019 · see citing works
Fractional Fourier transforms and their optical implementation: I
Mendlovic, D., & Ozaktas, H. M. (1993) · Journal of the Optical Society of America A · doi:10.1364/josaa.10.001875 · Cited by 848 · see citing works
Self-citationThe Fourier Integral: and certain of its Applications
Wiener, N. (1933) · Nature · doi:10.1038/132731b0 · Cited by 846 · see citing works
On Namias's Fractional Fourier Transforms
McBRIDE, A. C., & KERR, F. H. (1987) · IMA Journal of Applied Mathematics · doi:10.1093/imamat/39.2.159 · Cited by 597 · see citing works
Fourier transforms of fractional order and their optical interpretation
Özaktaş, H. M., & Mendlovic, D. (1993) · Optics Communications · doi:10.1016/0030-4018(93)90359-d · Cited by 279 · see citing works
Self-citationEigenvectors and functions of the discrete Fourier transform
Dickinson, B., & Steiglitz, K. (1982) · IEEE Transactions on Acoustics, Speech, and Signal Processing · doi:10.1109/tassp.1982.1163843 · Cited by 252 · see citing works
The wigner distribution function and its optical production
Bartelt, H. O., Brenner, K., & Lohmann, A. W. (1980) · Optics Communications · doi:10.1016/0030-4018(80)90308-9 · Cited by 242 · see citing works
Diffraction from a wavelet point of view
Onural, L. (1993) · Optics Letters · doi:10.1364/ol.18.000846 · Cited by 90 · see citing works
Self-citationCaola, M. J. (1991) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/24/19/004 · Cited by 70 · see citing works
An introduction to the angular Fourier transform
Almeida, L. B. (1993) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp.1993.319481 · Cited by 62 · see citing works
Self-Fourier objects and other self-transform objects
Lohmann, A. W., & Mendlovic, D. (1992) · Journal of the Optical Society of America A · doi:10.1364/josaa.9.002009 · Cited by 52 · see citing works
Self-citationGeneralized self-Fourier functions
Cincotti, G., Gori, F., & Santarsiero, M. (1992) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/25/20/001 · Cited by 49 · see citing works
Self Fourier functions and fractional Fourier transforms
Mendlovic, D., Özaktaş, H. M., & Lohmann, A. W. (1994) · Optics Communications · doi:10.1016/0030-4018(94)90289-5 · Cited by 39 · see citing works
Self-citationOptical-coordinate transformation methods and optical-interconnection architectures
Mendlovic, D., & Özaktaş, H. M. (1993) · Applied Optics · doi:10.1364/ao.32.005119 · Cited by 16 · see citing works
Self-citation
Flagged first, then by citations
Research works printed but not linked: 4
- [2]2. J. Wood and D. T. Barry, "Radon transformation of the Wigner spec- trum," in Advanced Signal Processing Algorithms, Ar- chitectures, and Implementations III, R T. Luk, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1770, 358-375 (1992);
- [9]9. D. Mendlovic, H. M. Ozaktas, and A. W Lohmann, "Fourier transforms of fractional order and their optical interpreta- tion," in Optical Computing, Vol. 7 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 127-130.
- [17]17. A. W Lohmann and B. H. Soffer, "Relationship between two transforms: Radon-Wigner and fractional Fourier," in An- nual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 109.
- [18]18. A. Rosenfeld and A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, San Diego, Calif., 1982), Vol. 1.
Cited by
Sorted by citations
The fractional Fourier transform and time-frequency representations
Almeida, L. B. (1994) · IEEE Transactions on Signal Processing · doi:10.1109/78.330368 · Cited by 1,924 · see citing works
Digital computation of the fractional Fourier transform
Ozaktas, H. M., Arikan, O., Kutay, M. A., & Bozdagt, G. (1996) · IEEE Transactions on Signal Processing · doi:10.1109/78.536672 · Cited by 1,187 · see citing works
Self-citationThe 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-citationDiscrete fractional Fourier-transform based on orthogonal projections
Pei, S.-C., Yeh, M.-H., & Tseng, C.-C. (1999) · IEEE Transactions on Signal Processing · doi:10.1109/78.757221 · Cited by 297 · see citing works
A convolution and product theorem for the fractional Fourier transform
Zayed, A. I. (1998) · IEEE Signal Processing Letters · doi:10.1109/97.664179 · Cited by 239 · see citing works
Improved discrete fractional Fourier transform
Pei, S.-C., & Yeh, M.-H. (1997) · Optics Letters · doi:10.1364/ol.22.001047 · Cited by 188 · see citing works
introduction to the fractional fourier transform and its applications
Ozaktas, H. M., Kutay, M. A., & Mendlovic, D. (1999) · Advances in Imaging and Electron Physics · doi:10.1016/s1076-5670(08)70272-6 · Cited by 164 · see citing works
Self-citationOzaktas, H. M., & Aytür, O. (1995) · Signal Processing · doi:10.1016/0165-1684(95)00076-p · Cited by 141 · see citing works
Self-citationThe discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform
Barker, L., Candan, Ç., Hakioglu, T., Kutay, M. A., & Ozaktas, H. M. (2000) · Journal of Physics A: Mathematical and General · doi:10.1088/0305-4470/33/11/304 · Cited by 76 · see citing works
Time delay estimation using fractional Fourier transform
Sharma, K. K., & Joshi, S. D. (2007) · Signal Processing · doi:10.1016/j.sigpro.2006.08.007 · Cited by 62 · see citing works
Ozaktas, H. M., Erkaya, N., & Kutay, M. A. (1996) · IEEE Signal Processing Letters · doi:10.1109/97.484211 · Cited by 62 · see citing works
Self-citationThe discrete fractional Fourier transform
Candan, C., Kutay, M. A., & Ozaktas, H. M. (1999) · 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258) · doi:10.1109/icassp.1999.756324 · Cited by 52 · see citing works
Nonseparable two-dimensional fractional Fourier transform
Sahin, A., Kutay, M. A., & Ozaktas, H. M. (1998) · Applied Optics · doi:10.1364/ao.37.005444 · Cited by 41 · see citing works
Self-citationPrajna, K., & Mukhopadhyay, C. K. (2020) · Journal of Nondestructive Evaluation · doi:10.1007/s10921-020-0658-6 · Cited by 38 · see citing works
Space–bandwidth-efficient realizations of linear systems
Kutay, M. A., Erden, M. F., Ozaktas, H. M., Arıkan, O., Güleryüz, Ö., & Candan, Ç. (1998) · Optics Letters · doi:10.1364/ol.23.001069 · Cited by 22 · see citing works
Self-citationHuang, D.-F., & Chen, B.-S. (2000) · Signal Processing · doi:10.1016/s0165-1684(00)00052-9 · Cited by 12 · see citing works
Fractional convolution, correlation theorem and its application in filter design
Feng, Q., & Wang, R.-B. (2020) · Signal, Image and Video Processing · doi:10.1007/s11760-019-01563-9 · Cited by 10 · see citing works
Kutay, M. A., Erden, M. F., Ozaktas, H. M., Arikan, O., Candan, C., & Guleryuz, O. (2002) · Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181) · doi:10.1109/icassp.1998.679603 · Cited by 8 · see citing works
Self-citationSolution and cost analysis of general multi-channel and multi-stage filtering circuits
Kutay, M. A., Ozaktas, H., Erden, M. F., Ozaktas, H. M., & Arikan, O. (2002) · Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (Cat. No.98TH8380) · doi:10.1109/tfsa.1998.721466 · Cited by 5 · see citing works
Self-citation
- Authors
- Ozaktas, H. M., Mendlovic, D., Onural, L., & Barshan, B.
- Affiliations
- Bilkent University (TR)
- Venue
- Journal of the Optical Society of America A
- Publisher
- Optica Publishing Group
- Published
- 1994-02-01
- Volume
- 11
- Issue
- 2
- Pages
- 547
- ISSN
- 1084-7529 (ISSN-L)
- 1520-8532
- OpenAlex
- W2169175428
- Semantic Scholar
- b63beec5b8ca3288a20b826c776cb9c8f1b8481c
- Cited by
- 483
- References
- 15
- Citations by year
- 2026: 3 · 2025: 5 · 2024: 6 · 2023: 4 · 2022: 8 · 2021: 4 · 2020: 7 · 2019: 10 · 2018: 10 · 2017: 11 · 2016: 12 · 2015: 20 · 2014: 18 · 2013: 20 · 2012: 16
- FWCI
- 60.64
- Type
- journal article
- Language
- en
- Open access
- green
- hdl.handle.net
- Locations
- Journal of the Optical Society of America A (publishedVersion)
- Bilkent University Institutional Repository (Bilkent University) (submittedVersion)
- Funders
- Bilkent Üniversitesi
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical Analysis and Transform Methods
- Physical Sciences › Computer Science › Signal Processing › Digital Filter Design and Implementation
- Physical Sciences › Engineering › Computational Mechanics › Sparse and Compressive Sensing Techniques
- Keywords
- fractional Fourier transform, wavelet transform, Wigner function, convolution, filtering, multiplexing, optical information processing, optical signal processing
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