Journal ArticleDOI
Eigenvalue and eigenvector decomposition of the discrete Fourier transform
J. McClellan,T. Parks +1 more
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The eigenvalues of a suitably normalized version of the discrete Fourier transform (DFT) are {1, -1,j, -j} and an eigenvector basis is constructed for the DFT.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.read more
Citations
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SVD representation of unitarily invariant matrices
TL;DR: In this article, it was shown that the forward-backward data matrix arising in linear prediction is bisymmetric, which can be used to establish algebraic invariance properties possessed by the singular vectors associated with that matrices's SVD representation.
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Generalized eigenvectors and fractionalization of offset DFTs and DCTs
Soo-Chang Pei,Jian-Jiun Ding +1 more
TL;DR: This paper derives the eigenvectors/eigenvalues of the offset DFT by convolution theorem, and can derive the close form eigenvector sets of theoffset DFT when a+b is an integer.
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Discrete fractional Fourier transform based on the eigenvectors of tridiagonal and nearly tridiagonal matrices
TL;DR: Simulation results show that the eigenvectors of matrix S better approximate samples of the Hermite-Gaussian functions than those of matrix T and moreover they have a shorter computation time due to the block diagonalization result, which can serve as a better basis for developing the DFRFT.
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Direct Batch Evaluation of Optimal Orthonormal Eigenvectors of the DFT Matrix
TL;DR: A direct technique for the collective (batch) evaluation of optimal Hermite-Gaussian-like eigenvectors of matrix F is contributed which is faster than the OPA and solves for the entire target modal matrix of F instead of the sequential generation of the eigenevectors.
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The discrete fractional Fourier transform and Harper's equation
TL;DR: In this paper, it was shown that the discrete fractional Fourier transform recovers the continuum fractional decomposition via a limiting process whereby inner products are preserved, and it is shown that this limiting process can be used to preserve the inner products of the discrete Fourier transformation.
References
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Journal ArticleDOI
The finite Fourier transform
TL;DR: In this paper, the finite Fourier transform of a finite sequence is defined and its elementary properties are developed, and the convolution and term-by-term product operations are defined and their equivalent operations in transform space.
Journal ArticleDOI
A short bibliography on the fast Fourier transform
TL;DR: A guided tour of the fast Fourier transform,” IEEE Spectrum (to be published).