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In mathematics, the Fourier transform is an integral transform that determines the frequency spectrum for a given waveform. Specifically, it transforms a function ƒ(t) describing the shape of a wave to a complex-valued function F(ν) describing the amplitude and phase of each frequency component.

Mathematically, the Fourier transform can be defined by the following formula:

${\displaystyle F(\nu )=\int _{-\infty }^{\infty }f(t)e^{-2\pi i\nu t}\,dt.}$

There are several different versions of this formula in common use, which give rise to slightly different functions F(ν). In addition, it is possible to avoid complex numbers by using the Fourier sine and cosine transforms. See also the list of Fourier-related transforms.

The Fourier transform is fundamental to the mathematical study of waves. As such, it is used extensively in physics and engineering, especially in signal processing, quantum mechanics, optics, and acoustics. In mathematics, Fourier transforms and Fourier series are the central objects of study in Fourier analysis, which can be considered a special case of abstract harmonic analysis.

In the study of waves, it is possible to describe any waveform or signal as a combination of simpler waves, each of which has a single frequency. For example, a sound wave can be described as a combination of different pitches, and a beam of light can be described as a combination of different spectral colors.

Mathematically, a single-frequency wave is just a sinusoidal function:

${\displaystyle f(t)=A\,\cos(2\pi \nu t+\phi )\!}$

Here A is the amplitude, ν is the frequency, and φ is the phase of the wave. A multi-frequency wave can be written as a sum or integral of simple sinusoidal waves:

${\displaystyle f(t)=\int _{0}^{\infty }A(\nu )\,\cos(2\pi \nu t+\phi (\nu ))\,d\nu \!}$

Here A and φ have become functions, since they depend on the frequency ν. The integral is necessary because the possible frequencies ν form a continuous spectrum. (In the case where ƒ is periodic, only a discrete set of frequencies is present, and the integral becomes a sum. The result is known as a Fourier series.)

For technical reasons, it works better to replace the cosines with complex exponentials:

${\displaystyle f(t)=\int _{-\infty }^{\infty }F(\nu )e^{2\pi i\nu t}\,d\nu \!}$

The complex-valued function F(ν) incorporates both the amplitude and the phase:

${\displaystyle F(\nu )=A(\nu )e^{i\phi (\nu )}\quad {\text{and}}\quad F(-\nu )=A(\nu )e^{-i\phi (\nu )}}$

The function F(ν) is called the Fourier transform of the function ƒ(t). Somewhat surprisingly, there is a simple formula for F in terms of ƒ:

${\displaystyle F(\nu )=\int _{-\infty }^{\infty }f(t)e^{-2\pi i\nu t}\,dt.\!}$

There are several common notations for the Fourier transform of a function ${\displaystyle f(t)\!}$, including ${\displaystyle F(\nu ),\!}$   ${\displaystyle {\hat {f}}(\nu ),}$  ${\displaystyle {\mathcal {F}}{\big \{}f(t){\big \}},}$   ${\displaystyle {\mathcal {F}}\{f\}(\nu ),}$   and ${\displaystyle ({\mathcal {F}}f)(\nu ).}$ This article uses the notation ${\displaystyle F(\nu )\!}$ throughout.

There are also several notable variations on the formula for the Fourier transform. While the form we have given is common in engineering, physicists prefer to use the angular frequency ${\displaystyle \omega =2\pi \nu \!}$. This leads to the following formula for the transform:

${\displaystyle F_{1}(\omega )=\int _{-\infty }^{\infty }f(t)e^{-i\omega t}\,dt}$

The function F₁(ω) is related to F(ν) by the following formulas:

${\displaystyle F(\nu )=F_{1}(2\pi \nu )\quad {\text{and}}\quad F_{1}(\omega )=F(\omega /2\pi )}$

Unfortunately, this convention leads to an awkward formula for ƒ in terms of F₁:

${\displaystyle f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F_{1}(\omega )e^{i\omega t}d\omega }$

To eliminate the asymmetric placement of the 2π, it is common to include a square root of 2π in both formulas:

${\displaystyle F_{2}(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(t)e^{-i\omega t}\,dt\quad {\text{and}}\quad f(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }F_{2}(\omega )e^{i\omega t}\,d\omega .}$

Here ${\displaystyle F_{2}(\omega )=F_{1}(\omega )/{\sqrt {2\pi }}}$. The Fourier transforms ${\displaystyle f(t)\mapsto F(\nu )}$ and ${\displaystyle f(t)\mapsto F_{2}(\omega )}$ are both unitary, while the transformation ${\displaystyle f(t)\mapsto F_{1}(\omega )}$ is not.

Let ƒ(t) be a function, either real- or complex-valued, defined for all real numbers t. The Fourier transform of ƒ is the function F(ν), defined by the following formula:

${\displaystyle F(\nu )=\int _{-\infty }^{\infty }f(t)e^{-2\pi i\nu t}\,dt.}$

In applications, the functions ƒ and F are thought of as two aspects of the same waveform: the function ƒ describes the wave over the time domain, while F describes the wave over the frequency domain.

There are several common conventions for defining the Fourier transform of a complex-valued Lebesgue integrable function f :R→C. One common definition is: real number ν.

When the independent variable t represents time (with SI unit of seconds), the transform variable ν  represents ordinary frequency (in hertz). Under suitable conditions, f can be reconstructed from F  by the inverse transform:

${\displaystyle f(t)=\int _{-\infty }^{\infty }F(\nu )\ e^{2\pi i\nu t}\,d\nu ,}$   for every real number t.

Other notations for ${\displaystyle F(\nu )\,}$ are:  ${\displaystyle {\hat {f}}(\nu ),}$  ${\displaystyle {\mathcal {F}}{\big \{}f(t){\big \}},}$  ${\displaystyle {\mathcal {F}}\{f\}(\nu ),}$ and ${\displaystyle ({\mathcal {F}}f)(\nu ).}$

The interpretation of the complex function ${\displaystyle F(\nu )\,}$ may be aided by expressing it in polar coordinate form:  ${\displaystyle F(\nu )=A(\nu )\ e^{i\phi (\nu )},\,}$   in terms of the two real functions ${\displaystyle A(\nu )\,}$ and ${\displaystyle \phi (\nu )\,}$, where:

${\displaystyle A(\nu )=|F(\nu )|,\,}$

is the amplitude and

${\displaystyle \phi (\nu )=\arg {\big (}F(\nu ){\big )},\,}$  

is the phase (see arg function).

Then the inverse transform can be written:

${\displaystyle f(t)=\int _{-\infty }^{\infty }A(\nu )\ e^{i(2\pi \nu t+\phi (\nu ))}\,d\nu ,}$

which is a recombination of all the frequency components of f (t ). Each component is a complex sinusoid of the form e ^2πiνt  whose amplitude is A (ν ) and whose initial phase angle (at t =0) is φ (ν ).

The Fourier transform is often written in terms of angular frequency:   ω = 2πν whose units are radians per second.

The substitution ν = ω/(2π) into the formulas above produces this convention:

${\displaystyle F(\omega )=\int _{-\infty }^{\infty }f(t)\ e^{-i\omega t}\,dt}$

${\displaystyle f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(\omega )\ e^{i\omega t}\,d\omega ,}$

which is also a bilateral Laplace transform evaluated at s=iω.

The 2π factor can be split evenly between the Fourier transform and the inverse, which leads to another popular convention:

${\displaystyle F(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(t)\ e^{-i\omega t}\,dt}$

${\displaystyle f(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }F(\omega )\ e^{i\omega t}\,d\omega .}$

This makes the transform a unitary one.

Variations of all three conventions can be created by conjugating the complex-exponential kernel of both the forward and the reverse transform. The signs must be opposites. Other than that, the choice is (again) a matter of convention.