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Eigenvalue and eigenvector decomposition of the discrete Fourier transform
Abstract
The principal results of this paper are listed as follows. 1) The eigenvalues of a suitably normalized version of the discrete Fourier transform (DFT) are{1, -1,j, -j}. 2) An eigenvector basis is constructed for the DFT. 3) The multiplicities of the eigenvalues are summarized for an N×N transform as follows.
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“This transform matrix is unitary since the eigenvalues of the DFT matrix have unit magnitude [38], [41].”
From The discrete fractional Fourier transform (2000) · cites this paper as [41]
In the citing paper
“Since the DFT matrix has only four distinct eigenvalues [41], the eigenvalues are in general degenerate so that the eigenvector set is not unique.”
From The discrete fractional Fourier transform (2000) · cites this paper as [41]
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Metadata
- Authors
- McClellan, J., & Parks, T.
- Affiliations
- Rice University (US)
- ORCID
- McClellan, J.
- Venue
- IEEE Transactions on Audio and Electroacoustics
- Published
- 1972-03
- Volume
- 20
- Issue
- 1
- Pages
- 66-74
- ISSN
- 0018-9278 (ISSN-L)
- OpenAlex
- W2102337953
- Cited by
- 257
- References
- 4
- Citations by year
- 2026: 3 · 2025: 5 · 2024: 5 · 2023: 32 · 2022: 5 · 2021: 8 · 2020: 4 · 2019: 9 · 2018: 13 · 2017: 8 · 2016: 3 · 2015: 6 · 2014: 11 · 2013: 11 · 2012: 14
- FWCI
- 0.54
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- IEEE Transactions on Audio and Electroacoustics (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Signal Processing › Digital Filter Design and Implementation
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical Analysis and Transform Methods
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Matrix Theory and Algorithms
- Keywords
- discrete Fourier transform, eigenvalue decomposition, eigenvalues, eigenvector basis, eigenvalue multiplicity, matrix eigenvalues
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