Eigenvalue and eigenvector decomposition of the discrete Fourier transform
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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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323 citations
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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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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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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