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An algorithm for the machine calculation of complex Fourier series
Open accessAbstract
An efficient method for the calculation of the interactions of a 2m factorial experiment was introduced by Yates and is widely known by his name.The generalization to 3m was given by Box et al. [1].Good [2] generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series.In their full generality, Good's methods are applicable to certain problems in which one must multiply an JV-vector by an JV X N matrix which can be factored into m sparse matrices, where m is proportional to log JV.This results in a procedure requiring a number of operations proportional to JV log JV rather than JV2.These methods are applied here to the calculation of complex Fourier series.They are useful in situations where the number of data points is, or can be chosen to be, a highly composite number.The algorithm is here derived and presented in a rather different form.Attention is given to the choice of JV.It is also shown how special advantage can be obtained in the use of a binary computer with JV = 2m and how the entire calculation can be performed within the array of JV data storage locations used for the given Fourier coefficients.Consider the problem of calculating the complex Fourier series
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The publication of the fast Fourier transform meant that Fourier transforms that previously took hours to compute could be calculated in seconds.
“This meant that Fourier transforms that previously took hours to compute could be calculated in seconds.”
From A brief history of arterial wave mechanics (2009) · cites this paper as [6]
The paper reports IBM 7094 computing times of 0.02–0.13 minutes—about 1–8 seconds—for three-dimensional Fourier sums (Abstract and Additional information; page not provided). It also shows that the algorithm requires fewer operations than straightforward calculation, but it does not report that comparable transforms previously took hours. (LLM, conf=0.85)
Experimental — accuracy not yet measuredUnlike the DFT that can be self-factored to obtain FFT algorithms, we are generally unable to factorize the DVM to obtain a self-contained and radix-2 factorization, leading to an FFT-like algorithm.
“Therefore, unlike the case of the DFT that can be self-factored to obtain FFT algorithms [19,20,21], we are generally unable to factorize the DVM to obtain a self-contained and radix-2 factorization, leading to an FFT-like algorithm.”
From Thiran Filters for Wideband DSP-Based Multi-Beam True Time Delay RF Sensing Applications (2024) · cites this paper as [19]
The rapid growth in the application of Fourier analysis to arterial mechanics coincides with the development of the digital computer and the publication of the fast Fourier transform.
“It is probably not a coincidence that the rapid growth in the application of Fourier analysis to arterial mechanics coincides with the development of the digital computer and the publication of the fast Fourier transform [6]”
From A brief history of arterial wave mechanics (2009) · cites this paper as [6]
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Metadata
- Authors
- Cooley, J. W., & Tukey, J. W.
- Venue
- Mathematics of Computation
- Publisher
- American Mathematical Society (AMS)
- Published
- 1965
- Volume
- 19
- Issue
- 90
- Pages
- 297-301
- ISSN
- 0025-5718 (ISSN-L)
- 1088-6842
- OpenAlex
- W2061171222
- Cited by
- 13,154
- References
- 2
- Citations by year
- 2026: 504 · 2025: 639 · 2024: 531 · 2023: 535 · 2022: 459 · 2021: 501 · 2020: 436 · 2019: 483 · 2018: 420 · 2017: 441 · 2016: 391 · 2015: 407 · 2014: 422 · 2013: 413 · 2012: 382
- FWCI
- 88.25
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Mathematics of Computation (publishedVersion)
- Funders
- Princeton University
- Army Research Office
- Topics
- Physical Sciences › Computer Science › Signal Processing › Digital Filter Design and Implementation
- Physical Sciences › Mathematics › Modeling and Simulation › Mathematical and Computational Methods
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical functions and polynomials
- Keywords
- complex Fourier series, sparse matrix factorization, fast matrix-vector multiplication
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