Closed-Form Orthogonal DFT Eigenvectors Generated by Complete Generalized Legendre Sequence

In this paper, we propose a new method for deriving the closed-form orthogonal discrete Fourier transform (DFT) eigenvectors of arbitrary length using the complete generalized Legendre sequence (CGLS). From the eigenvectors, we then develop a novel method for computing the DFT. By taking a specific...

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Bibliographic Details
Published in:IEEE transactions on circuits and systems. I, Regular papers Regular papers, 2008-12, Vol.55 (11), p.3469-3479
Main Authors: Pei, Soo-Chang, Wen, Chia-Chang, Ding, Jian-Jiun
Format: Article
Language:English
Subjects:
Algorithms
Circuits
Closed-form solution
Complete generalized Legendre sequence (CGLS)
Construction specifications
Councils
Cryptography
discrete Fourier transform (DFT) eigenvector
Discrete Fourier transforms
discrete fractional Fourier transform (DFRFT)
Eigenvalues and eigenfunctions
Eigenvectors
Encryption
Exact solutions
fast Fourier transform (FFT)
Fast Fourier transforms
Fourier transforms
Mathematical analysis
OFDM
Sampling methods
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Summary:In this paper, we propose a new method for deriving the closed-form orthogonal discrete Fourier transform (DFT) eigenvectors of arbitrary length using the complete generalized Legendre sequence (CGLS). From the eigenvectors, we then develop a novel method for computing the DFT. By taking a specific eigendecomposition to the DFT matrix, after proper arrangement, we can derive a new fast DFT algorithm with systematic construction of an arbitrary length that reduces the number of multiplications needed as compared with the existing fast algorithm. Moreover, we can also use the proposed CGLS-like DFT eigenvectors to define a new type of the discrete fractional Fourier transform, which is efficient in implementation and effective for encryption and OFDM.
ISSN:1549-8328
1558-0806
DOI:10.1109/TCSI.2008.925353