Journal ArticleDOI
Eigenvalue and eigenvector decomposition of the discrete Fourier transform
J. McClellan,T. Parks +1 more
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TLDR
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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The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform
TL;DR: In this article, the authors define a discrete fractional Fourier transform (FT) which is essentially the time-evolution operator of the discrete harmonic oscillator, and define its energy eigenfunctions as a discrete algebraic analogue of the Hermite-Gaussian functions.
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Discrete Fractional Fourier Transform Based on New Nearly Tridiagonal Commuting Matrices
TL;DR: A new nearly tridiagonal matrix is proposed, which commutes with the discrete Fourier transform (DFT) matrix and is shown to be DFT eigenvectors, which are more similar to the continuous Hermite-Gaussian functions than those developed before.
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On the multiangle centered discrete fractional Fourier transform
J.G. Vargas-Rubio,Balu Santhanam +1 more
TL;DR: This letter defines a DFRFT based on a centered version of the DFT (CDFRFT) using eigenvectors derived from the Gru/spl uml/nbaum tridiagonal commutor that serve as excellent discrete approximations to the Hermite-Gauss functions.
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On the application of uniform linear array bearing estimation techniques to uniform circular arrays
A.H. Tewfik,W. Hong +1 more
TL;DR: It is shown that the transformed sequence may be processed using a ROOT MUSIC based approach to estimate the elevations and azimuths of the observed sources.
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Eigenvalues and eigenvectors of generalized DFT, generalized DHT, DCT-IV and DST-IV matrices
TL;DR: The eigenvalues and eigenvectors of the generalized discrete Fourier transform, the GDHT, Hartley transform, DCT-IV, and DST-IV matrices, and the type-IV discrete sine transform matrices are investigated in a unified framework to demonstrate the effectiveness of fractional transforms.
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).