The Fourier transform is a generalization of thecomplexFourier series in the limit as. Replace the discrete with the continuous while letting. Then change the sum to anintegral, and the equations become

			(1)
			(2)

Here,

			(3)
			(4)

is called theforward () Fourier transform, and

			(5)
			(6)

is called theinverse () Fourier transform. The notation is introduced in Trott (2004, p. xxxiv), and and are sometimes also used to denote the Fourier transform and inverse Fourier transform, respectively (Krantz 1999, p. 202).

Note that some authors (especially physicists) prefer to write the transform in terms of angular frequency instead of the oscillation frequency. However, this destroys the symmetry, resulting in the transform pair

			(7)
			(8)
			(9)
			(10)

To restore the symmetry of the transforms, the convention

			(11)
			(12)
			(13)
			(14)

is sometimes used (Mathews and Walker 1970, p. 102).

In general, the Fourier transform pair may be defined using two arbitrary constants and as

			(15)
			(16)

The Fourier transform of a function is implemented theWolfram Language asFourierTransform[f,x,k], and different choices of and can be used by passing the optionalFourierParameters->a,b option. By default, theWolfram Language takesFourierParameters as. Unfortunately, a number of other conventions are in widespread use. For example, is used in modern physics, is used in pure mathematics and systems engineering, is used in probability theory for the computation of thecharacteristic function, is used in classical physics, and is used in signal processing. In this work, following Bracewell (1999, pp. 6-7),it is always assumed that and unless otherwise stated. This choice often results in greatly simplified transforms of common functions such as 1,, etc.

Since any function can be split up intoeven andodd portions and,

			(17)
			(18)

a Fourier transform can always be expressed in terms of theFourier cosine transform andFourier sine transform as

(19)

A function has a forward and inverse Fourier transform such that

(20)

provided that

1. exists.

2. There are a finite number of discontinuities.

3. The function has bounded variation. Asufficient weaker condition is fulfillment of theLipschitz condition

(Ramirez 1985, p. 29). The smoother a function (i.e., the larger the number of continuousderivatives), the more compact its Fourier transform.

The Fourier transform is linear, since if and have Fourier transforms and, then

			(21)
			(22)

Therefore,

			(23)
			(24)

The Fourier transform is also symmetric since implies.

Let denote theconvolution, then the transforms of convolutions of functions have particularly nice transforms,

			(25)
			(26)
			(27)
			(28)

The first of these is derived as follows:

			(29)
			(30)
			(31)
			(32)

where.

There is also a somewhat surprising and extremely important relationship between theautocorrelation and the Fourier transform known as theWiener-Khinchin theorem. Let, and denote thecomplex conjugate of, then the Fourier transform of theabsolute square of is given by

(33)

The Fourier transform of aderivative of a function is simply related to the transform of the function itself. Consider

(34)

Now useintegration by parts

(35)

with

			(36)
			(37)

and

			(38)
			(39)

then

(40)

The first term consists of an oscillating function times. But if the function is bounded so that

(41)

(as any physically significant signal must be), then the term vanishes, leaving

			(42)
			(43)

This process can be iterated for thethderivative to yield

(44)

The importantmodulation theorem of Fourier transforms allows to be expressed in terms of as follows,

			(45)
			(46)
			(47)
			(48)

Since thederivative of the Fourier transform is givenby

(49)

it follows that

(50)

Iterating gives the generalformula

			(51)
			(52)

Thevariance of a Fourier transform is

(53)

and it is true that

(54)

If has the Fourier transform, then the Fourier transform has the shift property

			(55)
			(56)

so has the Fourier transform

(57)

If has a Fourier transform, then the Fourier transform obeys a similarity theorem.

(58)

so has the Fourier transform

(59)

The "equivalent width" of a Fourier transform is

			(60)
			(61)

The "autocorrelation width" is

			(62)
			(63)

where denotes thecross-correlation of and and is thecomplex conjugate.

Any operation on which leaves itsarea unchanged leaves unchanged, since

(64)

The following table summarized some common Fourier transform pairs.

function
Fourier transform--1	1
Fourier transform--cosine
Fourier transform--delta function
Fourier transform--exponential function
Fourier transform--Gaussian
Fourier transform--Heaviside step function
Fourier transform--inverse function
Fourier transform--Lorentzian function
Fourier transform--ramp function
Fourier transform--sine

In two dimensions, the Fourier transform becomes

			(65)
			(66)

Similarly, the-dimensional Fourier transform can be defined for, by

			(67)
			(68)

See also

Explore with Wolfram|Alpha

More things to try:

• fourier transform
• Fourier transform 1
• Fourier transform cosx

References

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Fourier Transform."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/FourierTransform.html

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