A generalized Mobius transform and arithmetic Fourier transforms

Knockaert, Luc

doi:10.1109/78.330357

Cited by 7 publications

(2 citation statements)

References 6 publications

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“…In [8] and [9], Knockaert presented the theory of generalized Möbius transform and gave a general formulation, which was developed by the authors [4]. Meanwhile, the 1-D AFT derivation in [15] was directly extended to 2-D function (see [10], [12], [16]).…”

Section: §1 Introduction and Notationmentioning

confidence: 99%

Dirichlet characters, Gauss sums and arithmetic Fourier transforms

Jing

Liu²

2014

Appl. Math. J. Chin. Univ.

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In this paper, a general algorithm for the computation of the Fourier coefficients of 2π-periodic (continuous) functions is developed based on Dirichlet characters, Gauss sums and the generalized Möbius transform. It permits the direct extraction of the Fourier cosine and sine coefficients. Three special cases of our algorithm are presented. A VLSI architecture is presented and the error estimates are given. §1 Introduction and notationThe arithmetic Fourier transform (AFT) offers a convenient method, based on the construction of weighted averages, to calculate the Fourier coefficients of a complex-valued periodic function. It was discovered by Bruns [3] in 1903. Similar algorithms were studied by Tufts and Sadasiv [15] for the calculation of the Fourier coefficients of even periodic functions. This method was extended in [11] to calculate the Fourier coefficients of both the even and odd components of any periodic function.In [8] and [9], Knockaert presented the theory of generalized Möbius transform and gave a general formulation, which was developed by the authors [4]. Meanwhile, the 1-D AFT derivation in [15] was directly extended to 2-D function (see [10], [12], [16]). Kelley [7] and coworkers proposed a row-column algorithm in which the 2-D Fourier transform is decomposed into 1-D AFT. Atlas [2] developed the concept of two-dimensional AFT using the Bruns' method. In [5], Bruns's version of 1-D AFT is extended to the 2-D case. Schiff [14] and Hsu [6] used the Möbius inversion formulae to compute the inverse Z-transform. Furthermore, some VLSI architectures for the AFT are developed in [7] and [13].Has AFT been exhaustive?

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Section: §1 Introduction and Notationmentioning

confidence: 99%

Dirichlet characters, Gauss sums and arithmetic Fourier transforms

Jing

Liu²

2014

Appl. Math. J. Chin. Univ.

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“…Although the main and original motivation of the arithmetic algorithm was the computation of the Fourier Transform, further generalizations were performed and the arithmetic approach was utilized to calculate other transforms. Dr. Luc Knockaert of Department of Information Technology at Ghent University, Belgium, amplified the Bruns procedure, defining a generalized Möbius transform [18,19]. Moreover, four versions of the cosine transform was shaped in the arithmetic transform formalism [20].…”

Section: Introduction and Historical Backgroundmentioning

confidence: 99%

A Short Survey on Arithmetic Transforms and the Arithmetic Hartley Transform

Cintra

Oliveira

2004

Journal of Communication and Information Systems

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Arithmetic complexity has a main role in the performance of algorithms for spectrum evaluation. Arithmetic transform theory offers a method for computing trigonometrical transforms with minimal number of multiplications. In this paper, the proposed algorithms for the arithmetic Fourier transform are surveyed. A new arithmetic transform for computing the discrete Hartley transform is introduced: the Arithmetic Hartley transform. The interpolation process is shown to be the key element of the arithmetic transform theory.

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Eigenstructure and fractionalization of the quaternion discrete Fourier transform

Ribeiro

Lima

2020

Optik

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A generalized Mobius transform and arithmetic Fourier transforms

Cited by 7 publications

References 6 publications

Dirichlet characters, Gauss sums and arithmetic Fourier transforms

Dirichlet characters, Gauss sums and arithmetic Fourier transforms

A Short Survey on Arithmetic Transforms and the Arithmetic Hartley Transform

Eigenstructure and fractionalization of the quaternion discrete Fourier transform

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