The discrete fractional Fourier transform
Abstract
We propose and consolidate a definition of the discrete fractional Fourier transform which generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform (FRT) generalizes the continuous ordinary Fourier Transform. This definition is based on a particular set of eigenvectors of the DFT which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The fact that this definition satisfies all the desirable properties expected of the discrete FRT, supports our confidence that it will be accepted as the definitive definition of this transform.
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The S_e and S_o matrices have eigenvectors whose number of zero crossings range from 0 to the respective maximum.
“Therefore, we can show that the and matrices have eigenvectors whose number of zero crossings range from 0 to and from 0 to , respectively [48].”
From The discrete fractional Fourier transform (2000) · cites this paper as [48]
This eigenvalue-based procedure enables accurate determination of the number of zero crossings of an eigenvector without employing any means of counting.
“This eigenvalue-based procedure enables us to accurately determine the number of zero crossings of an eigenvector without employing any means of counting. [48].”
From The discrete fractional Fourier transform (2000) · cites this paper as [48]
The odd eigenvector derived from the eigenvector of the matrix with zero crossings is an eigenvector of the matrix with zero crossings [48].
“It can be further shown that the odd eigenvector derived from the eigenvector of with zero crossings is an eigenvector of with zero crossings [48].”
From The discrete fractional Fourier transform (2000) · cites this paper as [48]
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- Authors
- Candan, C., Kutay, M. A., & Ozaktas, H. M.
- Affiliations
- Bilkent University (TR)
- ORCID
- Kutay, M.A.
- Venue
- 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258)
- Publisher
- IEEE
- Published
- 1999
- Pages
- 1713-1716 vol.3
- ISSN
- 1520-6149 (ISSN-L)
- OpenAlex
- W2155549970
- Cited by
- 52
- References
- 54
- Citations by year
- 2026: 1 · 2024: 1 · 2023: 2 · 2022: 1 · 2021: 1 · 2020: 3 · 2019: 4 · 2017: 1 · 2016: 5 · 2015: 3 · 2014: 2 · 2013: 5 · 2012: 3
- FWCI
- 6.31
- Type
- conference paper
- Language
- en
- Open access
- closed
- Locations
- IEEE International Conference on Acoustics Speech and Signal Processing (publishedVersion)
- Bilkent University Institutional Repository (Bilkent University) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical Analysis and Transform Methods
- Physical Sciences › Computer Science › Signal Processing › Digital Filter Design and Implementation
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical functions and polynomials
- Keywords
- discrete fractional Fourier transform, discrete Fourier transform, fractional Fourier transform, Hermite-Gaussian functions
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