Multiplicity of fractional Fourier transforms and their relationships
Abstract
The multiplicity of the fractional Fourier transform (FRT), which is intrinsic in any fractional operator, has been claimed by several authors, but never systematically developed. The paper starts with a general FRT definition, based on eigenfunctions and eigenvalues of the ordinary Fourier transform, which allows us to generate all possible definitions. The multiplicity is due to different choices of both the eigenfunction and the eigenvalue classes. A main result, obtained by a generalized form of the sampling theorem, gives explicit relationships between the different FRTs.
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The definition in [38] and [39] corresponds to a completely distinct definition of the fractional Fourier transform.
“The definition in [38] and [39] corresponds to a completely distinct definition of the fractional Fourier transform [40].”
From The discrete fractional Fourier transform (2000) · cites this paper as [40]
Many other definitions of the discrete fractional Fourier transform exist, but they have several theoretical disadvantages; for example, the fast approximate fractional Fourier transform is not a unitary operator like the discrete transform defined in the paper.
“Many other definitions of the discrete fractional Fourier transform do exist [2,21,22,5], but they have several theoretical disadvantages. For example the fast approximate fractional Fourier transform that we have discussed before is not a unitary operator like the discrete transform as defined above is.”
From Computation of the fractional Fourier transform (2004) · cites this paper as [5]
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The discrete fractional Fourier transform
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“9 Other definitions of the DFRFT Many other definitions of the discrete fractional Fourier transform do exist [2,21,22,5], but they have several theoretical disadvantages.”
BackgroundThe discrete fractional Fourier transform
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Fractional Fourier analysis using the mobius inversion formula
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- Authors
- Cariolaro, G., Erseghe, T., Kraniauskas, P., & Laurenti, N.
- Affiliations
- University of Padua (IT)
- Venue
- IEEE Transactions on Signal Processing
- Published
- 2000
- Volume
- 48
- Issue
- 1
- Pages
- 227-241
- ISSN
- 1053-587X (ISSN-L)
- OpenAlex
- W2166959930
- Cited by
- 89
- References
- 21
- Citations by year
- 2026: 1 · 2025: 1 · 2024: 2 · 2023: 1 · 2022: 1 · 2021: 1 · 2020: 5 · 2019: 3 · 2018: 4 · 2017: 2 · 2016: 3 · 2015: 3 · 2014: 3 · 2013: 5 · 2012: 4
- FWCI
- 5.03
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- IEEE Transactions on Signal Processing (publishedVersion)
- Padua Research Archive (University of Padova) (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 › Modeling and Simulation › Fractional Differential Equations Solutions
- Keywords
- fractional Fourier transform, eigenvalues, generalized sampling theorem
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