Construct · 1 fields · 3 citations

Fourier transform: a standalone construct

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Fouriertransform1 CONSTRUCTFourier transformFOURIER · FREQUENCIES

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Mathematics

Decomposition of a function into its constituent frequencies

Joseph Fourier introduced the idea in his 1822 treatise Théorie analytique de la chaleur, developed from an 1807 memoir, where, to solve how heat spreads through a solid, he showed that a function could be written as a sum of sine and cosine waves. The Fourier transform generalises this: it takes a function of time or space and re-expresses it as a spread of frequencies, reporting how much of each sinusoid (a smooth, repeating wave) the original signal contains, so that a signal and its frequency content become two views of the same object. Because it turns differentiation and convolution (the blending of two signals) into ordinary multiplication, it converts many differential equations and filtering problems into algebra, which is why it underpins signal processing, image analysis, quantum mechanics, and the study of partial differential equations. The fast Fourier transform, an efficient computational method published by Cooley and Tukey in 1965, made the calculation practical on digital hardware and placed the transform at the centre of modern engineering.[1][2][3]

Unit
A linear integral transform mapping a function to its frequency spectrum
Recovery means
The inverse Fourier transform reconstructs the original function from its spectrum
Measured by
Computed analytically by integration, or numerically on sampled data via the discrete and fast Fourier transform

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Nearby constructs

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References

  1. Fourier, J. (1822). Théorie analytique de la chaleur. Firmin Didot, père et fils.
  2. Cooley, J. W., & Tukey, J. W. (1965). An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation, 19(90), 297–301. https://doi.org/10.1090/S0025-5718-1965-0178586-1
  3. Bracewell, R. N. (2000). The Fourier transform and its applications (3rd ed.). McGraw-Hill.

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