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Discrete Fourier transform in arbitrary dimensions by a generalized Beevers–Lipson algorithm

Discrete Fourier transform in arbitrary dimensions by a generalized Beevers–Lipson algorithm The Beevers–Lipson procedure was developed as an economical evaluation of Fourier maps in two‐ and three‐dimensional space. Straightforward generalization of this procedure towards a transformation in ‐dimensional space would lead to nested loops over the coordinates, respectively, and different computer code is required for each dimension. An algorithm is proposed based on the generalization of the Beevers–Lipson procedure towards transforms in ‐dimensional space that contains the dimension as a variable and that results in a single piece of computer code for arbitrary dimensions. The computational complexity is found to scale as , where N is the number of pixels in the map, and it is independent of the dimension of the transform. This procedure will find applications in the evaluation of Fourier maps of quasicrystals and other aperiodic crystals, and in the maximum‐entropy method for aperiodic crystals. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Crystallographica Section A: Foundations and Advances Wiley

Discrete Fourier transform in arbitrary dimensions by a generalized Beevers–Lipson algorithm

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Publisher
Wiley
Copyright
Copyright © 2000 Wiley Subscription Services, Inc., A Wiley Company
ISSN
0108-7673
eISSN
1600-5724
DOI
10.1107/S0108767300000532
Publisher site
See Article on Publisher Site

Abstract

The Beevers–Lipson procedure was developed as an economical evaluation of Fourier maps in two‐ and three‐dimensional space. Straightforward generalization of this procedure towards a transformation in ‐dimensional space would lead to nested loops over the coordinates, respectively, and different computer code is required for each dimension. An algorithm is proposed based on the generalization of the Beevers–Lipson procedure towards transforms in ‐dimensional space that contains the dimension as a variable and that results in a single piece of computer code for arbitrary dimensions. The computational complexity is found to scale as , where N is the number of pixels in the map, and it is independent of the dimension of the transform. This procedure will find applications in the evaluation of Fourier maps of quasicrystals and other aperiodic crystals, and in the maximum‐entropy method for aperiodic crystals.

Journal

Acta Crystallographica Section A: Foundations and AdvancesWiley

Published: May 1, 2000

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