# Fourier transform
## English
### Alternative forms

* FT (initialism)

### Etymology

Named after French mathematician and physicist Jean Baptiste Joseph Fourier, who initiated the study of what is now harmonic analysis.

### Noun

Fourier transform (plural Fourier transforms)
* (mathematical analysis, harmonic analysis, physics, electrical engineering) 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.
    Fourier transforms are not limited to acting on functions of time, but the domain of the original function is commonly called the time domain.
    The Fourier transform of a function of time is a complex function of frequency, whose magnitude (absolute value) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency.
** 2002, J. F. James, A Student's Guide to Fourier Transforms, 2nd edition, Cambridge University Press, page 116:Since a separate integration is needed to give each point of the transformed function, the process would become extremely tedious if it were to be attempted manually and many ingenious devices have been invented for preforming Fourier transforms mechanically, electrically, acoustically and optically.
** 2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, Lecture Notes in Mathematics 1859, page 1,
    The trigonometric sums of ${\mathcal {G}}^{F}$ are thus (up to a scalar) the Fourier transforms of the characteristic functions of the $G^{F}\!\!$-orbits of ${\mathcal {G}}^{F}$.
** 2012, David Brandwood, Fourier Transforms in Radar and Signal Processing, Artech House, 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 transform 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 $f:\mathbb {R} \to \mathbb {C}$. (The two are often defined together to highlight their connectedness.)
** One form of this definition pair is:
    ${\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi ix\xi }\,dx$
    $f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )\ e^{2\pi ix\xi }\,d\xi$,
    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

* integral transform

#### Derived terms

* inverse Fourier transform
* continuous Fourier transform (CFT)
* discrete Fourier transform (DFT)
* fast Fourier transform (FFT)
* Fourier-transform infrared spectroscopy (FTIR)

#### Translations
### Further reading

* Fourier analysis on Wikipedia.Wikipedia
* Harmonic analysis on Wikipedia.Wikipedia
* Fourier series on Wikipedia.Wikipedia
* Time–frequency analysis on Wikipedia.Wikipedia
* Wavelet transform on Wikipedia.Wikipedia