An algorithm for the machine calculation of complex Fourier series.

Cooley, J. W., & Tukey, J. W. (1965). An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation, 19(90), 297-301.

Type

Journal article

Intellectual base

Primary research

Year

1965

Volume

19

Issue

90

Pages

297-301

DOI registrant

American Mathematical Society

Link

https://doi.org/10.1090/s0025-5718-1965-0178586-1matched

Cited from

mathematics

Cited by

2 citations across 2 artifacts.

Primes

• Periodicity
  • … interval can be expanded as a sum of sines and cosines at integer multiples of the fundamental frequency, a result that subsequently generalized to the Fourier transform for non-periodic functions, the discrete Fourier transform for sampled signals, and the Fast Fourier Transform algorithm of Cooley and Tukey 1965

Domain-specific

• Fourier Transform
  • Second, convolution in one domain becomes pointwise multiplication in the other: the output of a linear time-invariant filter is the inverse transform of the product \(\hat{f}(\xi)\cdot\hat{h}(\xi)\), replacing an integral operation with arithmetic at each frequency — the foundation of spectral filtering, signal deconvolution, and efficient computation via the fast Fourier transform (Cooley and Tukey, 1965), which reduces the \(O(n^2)\) naive computation to \(O(n \log n)\)

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