Fourier transform

English

Alternative forms

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Etymology

Named after French mathematician and physicist Template:W, who initiated the study of what is now harmonic analysis.

Noun

1. Template:Lb A particular integral transform that when applied to a function of time (such as a signal), converts the function to one that plots the original function's frequency composition; the resultant function of such a conversion.

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   • Template:Quote-book
   • 2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, Template:W 1859, page 1,

     The trigonometric sums of ${𝒢}^F$ are thus (up to a scalar) the Fourier transforms of the characteristic functions of the $G^F$-orbits of ${𝒢}^F$.

   • 2012, David Brandwood, Fourier Transforms in Radar and Signal Processing, Template:W, 2nd Edition, page 1,

     The Fourier transform is a valuable theoretical technique, used widely in fields such as applied mathematics, statistics, physics, and engineering.

Usage notes

• Like the term Template:M itself, Fourier transform can mean either the integral operator that converts a function, or the function that is the end product of the conversion process.
• The Fourier transform of a function $f$ is traditionally denoted $\hat{f}$. Several other notations are also used.
• There are also several different conventions used when it comes to defining the Fourier transform and its inverse for an integrable function $f:ℝ\to ℂ$. (The two are often defined together to highlight their connectedness.)
  • One form of this definition pair is:

    $\hat{f}(\xi )=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)e^{-2\pi ix\xi }dx$

    $f(x)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\hat{f}(\xi )e^{2\pi ix\xi }d\xi$,

  where the exponent (including its sign) reflects a convention in electrical engineering to use $f(x)=e^{2\pi i{\xi }_{0}x}$ for a signal with initial phase 0 and frequency ${\xi }_0$.

Hypernyms

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Derived terms

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• Template:L (CFT)
• Template:L (DFT)
• Template:L (FFT)
• Template:L (FTIR)

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Further reading

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