Eigenvalue and eigenvector decomposition of the discrete Fourier transform
TL;DR: 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.
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TL;DR: This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions, and is exactly unitary, index additive, and reduces to the D FT for unit order.
Abstract: We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform.
604 citations
Cites background or methods from "Eigenvalue and eigenvector decompos..."
...For completeness, we present a short proof of this important result, which will be utilized in developments below (see also [41] for an alternative proof)....
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...This transform matrix is unitary since the eigenvalues of the DFT matrix have unit magnitude [38], [41]....
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...Since the DFT matrix has only four distinct eigenvalues [41], the eigenvalues are in general degenerate so that the eigenvector set...
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...The eigenvectors of the DFT matrix are either even or odd vectors [41]....
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TL;DR: In this paper, a factorization of finite quantum systems in terms of smaller subsystems, based on the Chinese remainder theorem, is studied, and the general formalism is applied to the case of angular momentum.
Abstract: Quantum systems with finite Hilbert space are considered, and phase-space methods like the Heisenberg–Weyl group, symplectic transformations and Wigner and Weyl functions are discussed. A factorization of such systems in terms of smaller subsystems, based on the Chinese remainder theorem, is studied. The general formalism is applied to the case of angular momentum. In this context, SU(2) coherent states are used for analytic representations. Links between the theory of finite quantum systems and other fields of research are discussed.
323 citations
TL;DR: The proposed DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT and will provide similar transform and rotational properties as those of continuous fractional Fourier transforms.
Abstract: The continuous fractional Fourier transform (FRFT) performs a spectrum rotation of signal in the time-frequency plane, and it becomes an important tool for time-varying signal analysis. A discrete fractional Fourier transform has been developed by Santhanam and McClellan (see ibid., vol.42, p.994-98, 1996) but its results do not match those of the corresponding continuous fractional Fourier transforms. We propose a new discrete fractional Fourier transform (DFRFT). The new DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT. To obtain DFT Hermite eigenvectors, two orthogonal projection methods are introduced. Thus, the new DFRFT will provide similar transform and rotational properties as those of continuous fractional Fourier transforms. Moreover, the relationship between FRFT and the proposed DFRFT has been established in the same way as the conventional DFT-to-continuous-Fourier transform.
291 citations
Cites background from "Eigenvalue and eigenvector decompos..."
...1 shows the FRFT of the rectangular window function [ for ; and , elsewhere] for various angles....
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...Thus, taking at both sides, we have (21) The proof is completed....
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...The real parts of the FRFT or DFRFT in this paper are plotted by solid lines, and the imaginary parts of the FRFT or DFRFT are indicated by dashed lines....
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TL;DR: A method is presented for computing an orthonormal set of eigenvectors for the discrete Fourier transform (DFT) based on a detailed analysis of the eigenstructure of a special matrix which commutes with the DFT.
Abstract: A method is presented for computing an orthonormal set of eigenvectors for the discrete Fourier transform (DFT). The technique is based on a detailed analysis of the eigenstructure of a special matrix which commutes with the DFT. It is also shown how fractional powers of the DFT can be efficiently computed, and possible applications to multiplexing and transform coding are suggested.
243 citations
Cites background or methods from "Eigenvalue and eigenvector decompos..."
...First, we recall some facts about the DFT matrix F in (1), The minimal polynomial of F is 7\4 -1 [1], [3] -[5]; thus F obeys the equation F4 = I (2) where I is the (N X N) identity matrix....
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...(1) Thus the methods of linear algebra and matrix theory may be applied to study the DFT. McClellan and Parks [1] suggested that the eigenstructure of F, being a natural linear algebraic way of decomposing the matrix, might be of computational interest as well....
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TL;DR: In this article, the authors discuss the discrete Fourier transform and point out some computational problems in complex analysis where it can be fruitfully applied, such as trigonometric interpolation, conjugate periodic functions, and numerical inversion of Laplace transforms.
Abstract: In this paper we discuss the discrete Fourier transform and point out some computational problems in (mainly) complex analysis where it can be fruitfully applied. We begin by describing the elementary properties of the transform and its efficient implementation, both in the one-dimensional and in the multi-dimensional case, by the reduction formulas of Cooley, Lewis, and Welch (IBM Res, paper, 1967).The following applications are then discussed: Calculation of Fourier coefficients using attenuation factors; solution of Symm’s integral equation in numerical conformal mapping; trigonometric interpolation; determination of conjugate periodic functions and their application to Theodorsen’s integral equation for the conformal mapping of simply and of doubly connected regions; determination of Laurent coefficients with applications to numerical differentiation, generating functions, and the numerical inversion of Laplace transforms; determination of the “density” of the zeros of high degree polynomials. We then...
228 citations
Cites background from "Eigenvalue and eigenvector decompos..."
...…approaches to fast Fourier transforms are based on a factoring of the matrix representing the Fourier operator (Theilheimer (1969), Kahaner (1970), McClellan and Parks (1972)), or on determining remainders in the division of polynomials (Fiduccia (1972), Aho, Hopcroft and Ullmann (1974), Kahaner…...
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References
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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.
Abstract: The finite Fourier transform of a finite sequence is defined and its elementary properties are developed. The convolution and term-by-term product operations are defined and their equivalent operations in transform space are given. A discussion of the transforms of stretched and sampled functions leads to a sampling theorem for finite sequences. Finally, these results are used to give a simple derivation of the fast Fourier transform algorithm.
165 citations
TL;DR: A guided tour of the fast Fourier transform,” IEEE Spectrum (to be published).
Abstract: 166 L. E. Alsop and A. A. Nowroozi, “Fast Fourier analysis,” J. Geophys. Res., vol. 71, pp. 5482-5483, November 15, 1966. €3. Andrews, “A high-speed algorithm for the computer generation of Fourier transforms,” IEEE Trans. Computers (Short Notes), vol. C-17, pp. 373.375, April 1968. J. S . Bailey, “A fast Fourier transform without multiplications,” Proc. Symp. on Computer Processing in Communications, vol. 19, MKI Symposia Ser. New York: Polytechnic Press, 1969. V. Benignus, “Estimation of the coherence spectrum and its confidence interval using the fast Fourier transform,” this issue, pp. 145-150. G. D. Bergland, “The fast Fourier transform recursive equations for arbitrary length records,” Math. Computation, vol. 21, pp, 236-238, April 1967. -9 “A fast Fourier transform algorithm using base eight iterations,” Math. Computation, vol. 22, pp. 275-279, April 1968. -, “A fast Fourier transform algorithm for realvalued series,” Commun. A C M , vol. 11, pp. 703--710, October 1968. -, “A radix-eight fast Fourier transform subroutine for real-valued series,” this issue, pp. 138144. -, “A guided tour of the fast Fourier transform,” IEEE Spectrum (to be published). “Fast Fourier transform hardware implementations. I. An overview. 11. A survey,’’ this issue,
35 citations