In mathematics, the continuous Fourier transform is a certain linear operator that maps functions to other functions. Loosely speaking, the Fourier transform decomposes a function into a continuous spectrum of its frequency components, and the inverse transform synthesizes a function from its spectrum of frequency components. In mathematical physics, the Fourier transform of a signal $x(t)\backslash ,$ can be thought of as that signal in the "frequency domain." This is similar to the basic idea of the various other Fourier transforms including the Fourier series of a periodic function. (See also fractional Fourier transform for a generalization.)

Definition

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Suppose $x\backslash ,$ is a complex-valued Lebesgue integrable function. The Fourier transform to the frequency domain, $\backslash omega\backslash ,$, is given by the function:

$X(\backslash omega)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\backslash infty\; x(t)\; e^\{-\; i\backslash omega\; t\}\backslash ,dt$,   for every real number $\backslash omega\; \backslash ,$.

Other notations for this same function are:  $\backslash hat\{x\}(\backslash omega)\backslash ,$  and  $\backslash mathcal\{F\}\backslash \{x\backslash \}(\backslash omega)\backslash ,$.  The function is complex-valued in general.   ($i\backslash ,$, of course, represents the imaginary unit.)

If $X(\backslash omega)\backslash ,$ is defined as above, and $x(t)\backslash ,$ is sufficiently smooth, then it can be reconstructed by the inverse transform:

$x(t)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; X(\backslash omega)\; e^\{\; i\backslash omega\; t\}\backslash ,d\backslash omega$,   for every real number $t\; \backslash ,$.

The interpretation of $X(\backslash omega)\backslash ,$ is aided by expressing it in polar coordinate form, $X(\backslash omega)\; =\; A(\backslash omega\; )e^\{i\; \backslash phi\; (\backslash omega\; )\}\; \backslash ,$, where:

$A(\backslash omega\; )\; =\; |X(\backslash omega)|\; \backslash ,$ the amplitude

$\backslash phi\; (\backslash omega\; )\; =\; \backslash angle\; X(\backslash omega)\; \backslash ,$ the phase

Then the inverse transform can be written:

$x(t)\; =\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash frac\{A(\backslash omega)\}\{\backslash sqrt\{2\backslash pi\}\}\backslash cdot\; e^\{\; i(\backslash omega\; t\; +\backslash phi\; (\backslash omega\; ))\}\backslash ,d\backslash omega$

which is a recombination of all the frequency components of $x(t)\backslash ,$. Each component is a complex sinusoid of the form $e^\{\; i\backslash omega\; t\}\backslash ,$ whose amplitude is proportional to $A(\backslash omega)\backslash ,$ and whose initial phase (at t = 0) is $\backslash phi\; (\backslash omega\; )\backslash ,$.

Normalization factors and alternative forms

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The factors $1\backslash over\backslash sqrt\{2\backslash pi\}$ before each integral ensure that there is no net change in amplitude when one transforms from one domain to the other and back. A necessary and sufficient condition is that the product of the factors be $1\; \backslash over\; 2\; \backslash pi$. When they are chosen to be equal, the transform is referred to as unitary. A common non-unitary convention is shown here:

$X(\backslash omega)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; x(t)\; e^\{-\; i\backslash omega\; t\}\backslash ,dt$

$x(t)\; =\; \backslash frac\{1\}\{2\backslash pi\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; X(\backslash omega)\; e^\{\; i\backslash omega\; t\}\backslash ,d\backslash omega$

As a rule of thumb, mathematicians generally prefer the unitary transform (for symmetry reasons), engineers commonly use the non-unitary form (as a special case of the bilateral Laplace transform), and physicists use either convention depending on the application.

Yet another popular form is a transform to or from the domain of ordinary frequency, $f\; =\; \backslash frac\{\backslash omega\}\{2\backslash pi\}\backslash ,$.  In that case, the necessary product of the factors is just 1, so a unitary transform is the obvious choice:

$X(f)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; x(t)\; e^\{-\; 2\backslash pi\; i\; f\; t\}\backslash ,dt$

$x(t)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; X(f)\; e^\{2\backslash pi\; i\; f\; t\}\backslash ,df$

This form is commonly used in applications related to signal processing and communications systems.

And variations of all three forms 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.

In many (but not all) applications of the CFT, the independent variable t represents time (with SI unit of seconds), while in mathematics this is rarely the case. In these situations, the transform variable ω represents the angular frequency (in radians per second), whereas f represents ordinary frequency (in hertz).

Summary of popular forms of the Fourier transform

{| border="1" cellspacing="0" cellpadding="12" angular
frequency
$\backslash omega\; \backslash ,$

(rad/s) unitary $X\_1(\backslash omega)\; \backslash equiv\; \backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; x(t)\; \backslash \; e^\{-i\; \backslash omega\; t\}\backslash ,\; dt\; \backslash \; =\; \backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\}\; X\_2(\backslash omega)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\}\; X\_3(\backslash frac\{\backslash omega\}\{2\; \backslash pi\})\backslash ,$
$x(t)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; X\_1(\backslash omega)\; \backslash \; e^\{i\; \backslash omega\; t\}\backslash ,\; d\; \backslash omega\; \backslash$ non-unitary $X\_2(\backslash omega)\; \backslash equiv\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; x(t)\; \backslash \; e^\{-i\; \backslash omega\; t\}\; \backslash \; dt\; \backslash \; =\; \backslash sqrt\{2\; \backslash pi\}\backslash \; X\_1(\backslash omega)\; =\; X\_3(\backslash frac\{\backslash omega\}\{2\; \backslash pi\})\backslash ,$
$x(t)\; =\; \backslash frac\{1\}\{2\; \backslash pi\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; X\_2(\backslash omega)\; \backslash \; e^\{i\; \backslash omega\; t\}\; \backslash \; d\; \backslash omega\; \backslash$

ordinary
frequency
$f\; \backslash ,$
(hertz) unitary $X\_3(f)\; \backslash equiv\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; x(t)\; \backslash \; e^\{-2\; \backslash pi\; i\; f\; t\}\; \backslash \; dt\; \backslash \; =\; \backslash sqrt\{2\; \backslash pi\}\backslash \; X\_1(2\; \backslash pi\; f)\; =\; X\_2(2\; \backslash pi\; f)\backslash ,$
$x(t)\; =\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; X\_3(f)\; \backslash \; e^\{2\; \backslash pi\; i\; f\; t\}\backslash ,\; df\; \backslash$

Generalization

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There are several ways to define the Fourier transform pair. The "forward" and "inverse" transforms are always defined so that the operation of both transforms in either order on a function will return the original function. In other words, the composition of the transform pair is defined to be the identity transformation. Using two arbitrary real constants $a$ and $b$, the most general definition of the forward 1-dimensional Fourier transform is given by:

$X(\backslash omega)\; =\; \backslash sqrt\{\backslash frac\{|b|\}\{(2\; \backslash pi)^\{1-a\}\}\}\; \backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; x(t)\; e^\{-i\; b\; \backslash omega\; t\}\; \backslash ,\; dt$

and the inverse is given by:

$x(t)\; =\; \backslash sqrt\{\backslash frac\{|b|\}\{(2\; \backslash pi)^\{1+a\}\}\}\; \backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; X(\backslash omega)\; e^\{i\; b\; \backslash omega\; t\}\; \backslash ,\; d\backslash omega$

Note that the transform definitions are symmetric; they can be reversed by simply changing the signs of a and b.

The convention adopted in this article is $(a,b)\; =\; (0,1)$. The choice of a and b is usually chosen so that it is geared towards the context in which the transform pairs are being used. The non-unitary convention above is $(a,b)=(1,1)$. Another very common definition is $(a,b)=(0,2\backslash pi)$ which is often used in signal processing applications. In this case, the angular frequency $\backslash omega$ becomes ordinary frequency f. If f (or ω) and t carry units, then their product must be dimensionless. For example, t may be in units of time, specifically seconds, and f (or ω) would be in hertz (or radian/s).

Properties

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See also the "Table of important Fourier transforms" section below for other properties of the continuous Fourier transform.

Completeness

If we define the Fourier transform $\backslash mathcal\{F\}$ in this way on the set of complex-valued functions on the line with compact support and extend by continuity to the Hilbert space of square-integrable functions, then it is a unitary operator

$\backslash mathcal\{F\}:L^2(\backslash mathbb\{R\})\backslash rightarrow\; L^2(\backslash mathbb\{R\}).$

Moreover,

$\backslash mathcal\{F\}^2\; f(x)=f(-x),\backslash quad\backslash mbox\{and\}\backslash quad\backslash mathcal\{F\}^*=\backslash mathcal\{F\}^\{-1\}=\backslash mathcal\{F\}^3.$

Note that in this relation, $\backslash mathcal\{F\}^*$ refers to adjoint of the Fourier Transform operator.

Extensions

The Fourier transform can also be extended to integrable functions

$f:\; \backslash ,\; \backslash mathbb\{R\}^n\; \backslash to\; \backslash mathbb\{C\}.$

In this case the definition usually appears as

$\backslash mathcal\{F\}\backslash \{f\backslash \}(w)\; \backslash equiv\; \backslash int\_\{\backslash R^n\}\; f(x)e^\{-i\backslash omega\backslash cdot\; x\}\backslash ,dx.$

Where $\backslash omega\backslash in\; \backslash mathbb\{R\}^n$ and $\backslash omega\; \backslash cdot\; x$ is the inner product of the two vectors $\backslash omega$ and $x$. All the above properties and formulas remain valid.

Again one may also use this to define the continuous Fourier transform for compactly supported smooth funcitions, which are dense in $L^2(\backslash mathbb\{R\}^n).$ Parseval's theorem then allows us to extend the definition of the Fourier transform to functions on $L^2(\backslash mathbb\{R\}^n)$ by continuity.

Unfortunately, further extensions become more technical. One may use the Hausdorff-Young inequality inequality to define the Fourier transform for $f\backslash in\; L^p(\backslash mathbb\{R\}^n)$ for $1\backslash leq\; p\backslash leq\; 2$. The Fourier transform of functions in $L^p$ for the range

The Plancherel theorem and Parseval's theorem

If $f(t)$ and $g(t)$ are square-integrable and $F(\backslash omega)$ and $G(\backslash omega)$ are their Fourier transforms, then we have the Parseval's theorem:

$\backslash int\_\{-\backslash infty\}^\backslash infty\; f(t)\; g^*(t)\; \backslash ,\; dt\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; F(\backslash omega)\; G^*(\backslash omega)\; \backslash ,\; d\backslash omega,$

where the asterisk * denotes complex conjugation. Therefore, the Fourier transformation yields an isometric automorphism of the Hilbert space L²(R).

The Plancherel theorem, a special case of the Parseval's theorem, states that

$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash left|\; x(t)\; \backslash right|^2\; dt\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash left|\; X(\backslash omega)\; \backslash right|^2\; d\backslash omega.$

This theorem is usually interpreted as asserting the unitary property of the Fourier transform. See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.

Localization property

As a rule of thumb: the more concentrated $f(t)$ is, the more spread out is $F(\backslash omega)$. Therefore a function which equals its Fourier transform strikes a precise balance between being concentrated and being spread out. It is easy in theory to construct examples of such functions (called self-dual functions) because the Fourier transform has order 4 (that is, iterating it four times on a function returns the original function). The sum of the four iterated Fourier transforms of any function will be self-dual. There are also some explicit examples of self-dual functions, the most important being constant multiples of the function

$f(t)\; =\; \backslash exp\; \backslash left(\; \backslash frac\{-t^2\}\{2\}\; \backslash right).$

This function is related to Gaussian distributions, and in fact, is an eigenfunction of the Fourier transform operators. The Fourier transform also translates between smoothness and decay: if $f(t)$ is several times differentiable, then $F(\backslash omega)$ decays rapidly towards zero for $s\; \backslash to\; \backslash plusmn\; \backslash infin$.

This can be more quantitatively expressed as follows. Suppose $f(t)$ and $F(\backslash omega)$ are a Fourier transform pair. Without loss of generality, we can assume that $f(t)$ is normalized:

$\backslash int\_\{-\backslash infty\}^\backslash infty\; f(t)f^*(t)\backslash ,dt=1.$

It follows from Parseval's theorem that F(ω) is also normalized. If we define the expected value of a function A(t) as:

$\backslash langle\; A\backslash rangle\; \backslash equiv\; \backslash int\_\{-\backslash infty\}^\backslash infty\; A(t)f(t)f^*(t)\backslash ,dt$

and the expectation value of a function $B(\backslash omega)$ as:

$\backslash langle\; B\backslash rangle\; \backslash equiv\; \backslash int\_\{-\backslash infty\}^\backslash infty\; B(\backslash omega)F(\backslash omega)F^*(\backslash omega)\backslash ,d\backslash omega$

and then define the variance of $A(t)$ as:

$\backslash Delta^2\; A\backslash equiv\backslash langle\; A^2-\backslash langle\; A\backslash rangle\; ^2\backslash rangle$

and similarly for the variance of $B(\backslash omega)$, then it can be shown that

$\backslash Delta\; t\; \backslash Delta\; \backslash omega\; \backslash ge\; \backslash frac\{1\}\{2\}.$

The most famous practical example of this property is found in quantum mechanics. The momentum and position wave functions are Fourier transform pairs to within a factor of $h\; \backslash over\; 2\; \backslash pi$ and are normalized to unity. The above expression then becomes a statement of the Heisenberg uncertainty principle.

Analysis of differential equations

Fourier transforms, and the closely related Laplace transforms are widely used in solving differential equations. The Fourier transform is compatible with differentiation in the following sense: if f(t) is a differentiable function with Fourier transform F(ω), then the Fourier transform of its derivative is given by iω F(ω). This can be used to transform differential equations into algebraic equations. Note that this technique only applies to problems whose domain is the whole set of real numbers. By extending the Fourier transform to functions of several variables (as outlined below), partial differential equations with domain Rⁿ can also be translated into algebraic equations.

Convolution theorem

Main article: Convolution theorem

The Fourier transform translates between convolution and multiplication of functions. If $h(t)$ and $x(t)$ are integrable functions with Fourier transforms $H(\backslash omega)$ and $X(\backslash omega)$, respectively, and if the convolution of $h$ and $x$ exists and is integrable, then the Fourier transform of the convolution is given by the product of the Fourier transforms $H(\backslash omega)\; X(\backslash omega)$ (possibly multiplied by a constant factor depending on the Fourier normalization convention).

In the current normalization convention, this means that if

$y(t)\; =\; h(t)*x(t)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; h(\backslash tau)x(t\; -\; \backslash tau)\backslash ,d\backslash tau$

then

$Y(\backslash omega)\; =\; \backslash sqrt\{2\backslash pi\}\backslash cdot\; H(\backslash omega)X(\backslash omega).\backslash ,$

In LTI system theory, it is common to interpret $h(t)$ as the impulse response of a linear, time-invariant system with input $x(t)$ and output $y(t)$, since substituting the unit impulse for $x(t)$ yields $y(t)=h(t)$. In this case, $H(\backslash omega)$ represents the frequency response of the system.

Conversely, if $x(t)$ can be decomposed as the product of two other functions $p(t)$ and $q(t)$ such that their product $p(t)\; q(t)$ is integrable, then the Fourier transform of this product is given by the convolution of the respective Fourier transforms $P(\backslash omega)$ and $Q(\backslash omega)$, again with a constant scaling factor.

In the current normalization convention, this means that if

$x(t)\; =\; p(t)\; q(t)\backslash ,$

then

$X(\backslash omega)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash bigg(\; P(\backslash omega)\; *\; Q(\backslash omega)\; \backslash bigg)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\backslash infty\; P(\backslash alpha)Q(\backslash omega\; -\; \backslash alpha)\backslash ,d\backslash alpha.$

Cross-correlation theorem

In an analogous manner, it can be shown that if $h(t)$ is the cross-correlation of $f(t)$ and $g(t)$:

$h(t)=(f\backslash star\; g)(t)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; f^*(\backslash tau)\backslash ,g(t+\backslash tau)\backslash ,d\backslash tau$

then the Fourier transform of $h(t)$ is:

$H(\backslash omega)\; =\; \backslash sqrt\{2\backslash pi\}\backslash ,F^*(\backslash omega)\backslash ,G(\backslash omega)$

where capital letters are again used to signify the Fourier transform.

Tempered distributions

The most general and natural context for studying the continuous Fourier transform is given by the tempered distributions; these include all the integrable functions mentioned above and have the added advantage that the Fourier transform of any tempered distribution is again a tempered distribution and the rule for the inverse of the Fourier transform is universally valid. Furthermore, the useful Dirac delta is a tempered distribution but not a function; its Fourier transform is the constant function $1/\backslash sqrt\{2\backslash pi\}$. Distributions can be differentiated and the above mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.

Extension to higher dimensions

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The Fourier transform can be extended to an N-dimensional space in a straightforward manner. If f(x) is a function of an N-dimensional vector x in the space, and k is the corresponding vector in the transform space (sometimes called the wavevector), then

$F(\backslash mathbf\{k\})=$

\left(\frac{1}{\sqrt{2\pi}}\right)^N \int_{\mathbb{R}^N} f (\mathbf{x})\,e^{-i\,\mathbf{k} \cdot \mathbf{x}}\,d\mathbf{x}

where dx is an N-dimensional infinitesimal volume element in the space and the product in the exponential is the dot product. Thinking of the function $f(x)=1$ as a tempered distribution one can derive the relationship:

$\backslash delta(\backslash mathbf\{k\})=\backslash left(\backslash frac\{1\}\{2\backslash pi\}\backslash right)^N$

\int_{\mathbb{R}^N} e^{\pm i\,\mathbf{k} \cdot \mathbf{x}}\,d\mathbf{x}

as tempered distributions .This yields the inverse transform:

$f(\backslash mathbf\{x\})=$

\left(\frac{1}{\sqrt{2\pi}}\right)^N \int_{\mathbb{R}^N} F (\mathbf{k})\,e^{+i\,\mathbf{k} \cdot \mathbf{x}}\,d\mathbf{k}

Table of important Fourier transforms

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The following table records some important Fourier transforms. $G$ and $H$ denote Fourier transforms of $g(t)$ and $h(t)$, respectively. $g$ and $h$ may be integrable functions or tempered distributions. Note that the two most common unitary conventions are included.

Functional relationships

	Signal	Fourier transform unitary, angular frequency	Fourier transform unitary, ordinary frequency	Remarks
	$g(t)\backslash !\backslash equiv\backslash !$ $\backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\backslash !\backslash !G(\backslash omega)\; e^\{i\; \backslash omega\; t\}\; d\; \backslash omega\; \backslash ,$	$G(\backslash omega)\backslash !\backslash equiv\backslash !$ $\backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\backslash !\backslash !g(t)\; e^\{-i\; \backslash omega\; t\}\; dt\; \backslash ,$	$G(f)\backslash !\backslash equiv$ $\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\backslash !\backslash !g(t)\; e^\{-i\; 2\backslash pi\; f\; t\}\; dt\; \backslash ,$
1	$a\backslash cdot\; g(t)\; +\; b\backslash cdot\; h(t)\backslash ,$	$a\backslash cdot\; G(\backslash omega)\; +\; b\backslash cdot\; H(\backslash omega)\backslash ,$	$a\backslash cdot\; G(f)\; +\; b\backslash cdot\; H(f)\backslash ,$	Linearity
2	$g(t\; -\; a)\backslash ,$	$e^\{-\; i\; a\; \backslash omega\}\; G(\backslash omega)\backslash ,$	$e^\{-\; i\; 2\backslash pi\; a\; f\}\; G(f)\backslash ,$	Shift in time domain
3	$e^\{\; iat\}\; g(t)\backslash ,$	$G(\backslash omega\; -\; a)\backslash ,$	$G\; \backslash left(f\; -\; \backslash frac\{a\}\{2\backslash pi\}\backslash right)\backslash ,$	Shift in frequency domain, dual of 2
4	$g(a\; t)\backslash ,$	\frac{1}{ >a	\frac{1}{ >a	>a
5	$G(t)\backslash ,$	$g(-\backslash omega)\backslash ,$	$g(-f)\backslash ,$	Duality property of the Fourier transform. Results from swapping "dummy" variables of $t\; \backslash ,$ and $\backslash omega\; \backslash ,$.
6	$\backslash frac\{d^n\; g(t)\}\{dt^n\}\backslash ,$	$(i\backslash omega)^n\; G(\backslash omega)\backslash ,$	$(i\; 2\backslash pi\; f)^n\; G(f)\backslash ,$	Generalized derivative property of the Fourier transform
7	$t^n\; g(t)\backslash ,$	$i^n\; \backslash frac\{d^n\; G(\backslash omega)\}\{d\backslash omega^n\}\backslash ,$	$\backslash left\; (\backslash frac\{i\}\{2\backslash pi\}\backslash right)^n\; \backslash frac\{d^n\; G(f)\}\{df^n\}\backslash ,$	This is the dual to 6
8	$(g\; *\; h)(t)\backslash ,$	$\backslash sqrt\{2\backslash pi\}\; G(\backslash omega)\; H(\backslash omega)\backslash ,$	$G(f)\; H(f)\backslash ,$	$g\; *\; h\backslash ,$ denotes the convolution of $g\backslash ,$ and $h\backslash ,$ — this rule is the convolution theorem
9	$g(t)\; h(t)\backslash ,$	$(G\; *\; H)(\backslash omega)\; \backslash over\; \backslash sqrt\{2\backslash pi\}\backslash ,$	$(G\; *\; H)(f)\backslash ,$	This is the dual of 8

Square-integrable functions

  $\backslash cdot\; \backslash \; \backslash sqrt\{1\; -\; \backslash omega^2\}\; \backslash mathrm\{rect\}\; \backslash left(\; \backslash frac\{\backslash omega\}\{2\}\; \backslash right)$  $\backslash cdot\; \backslash \; \backslash sqrt\{1\; -\; 4\; \backslash pi^2\; f^2\}\; \backslash mathrm\{rect\}\; (\; \backslash pi\; f\; )$

	Signal	Fourier transform unitary, angular frequency	Fourier transform unitary, ordinary frequency	Remarks
	$g(t)\backslash !\backslash equiv\backslash !$ $\backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\backslash !\backslash !G(\backslash omega)\; e^\{i\; \backslash omega\; t\}\; d\; \backslash omega\; \backslash ,$	$G(\backslash omega)\backslash !\backslash equiv\backslash !$ $\backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\backslash !\backslash !g(t)\; e^\{-i\; \backslash omega\; t\}\; dt\; \backslash ,$	$G(f)\backslash !\backslash equiv$ $\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\backslash !\backslash !g(t)\; e^\{-i\; 2\backslash pi\; f\; t\}\; dt\; \backslash ,$
10	$\backslash mathrm\{rect\}(a\; t)\; \backslash ,$	$\backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\; a^2\}\}\backslash cdot\; \backslash mathrm\{sinc\}\backslash left(\backslash frac\{\backslash omega\}\{2\backslash pi\; a\}\backslash right)$	\frac{1}{ >a	The rectangular pulse and the normalized sinc function
11	$\backslash mathrm\{sinc\}(a\; t)\backslash ,$	$\backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\; a^2\}\}\backslash cdot\; \backslash mathrm\{rect\}\backslash left(\backslash frac\{\backslash omega\}\{2\; \backslash pi\; a\}\backslash right)$	\frac{1}{ >a	Dual of rule 10. The rectangular function is an idealized low-pass filter, and the sinc function is the non-causal impulse response of such a filter.
12	$\backslash mathrm\{sinc\}^2\; (a\; t)\; \backslash ,$	$\backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\; a^2\}\}\backslash cdot\; \backslash mathrm\{tri\}\; \backslash left(\; \backslash frac\{\backslash omega\}\{2\backslash pi\; a\}\; \backslash right)$	\frac{1}{ >a	tri is the triangular function
13	$\backslash mathrm\{tri\}\; (a\; t)\; \backslash ,$	$\backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\; a^2\}\}\; \backslash cdot\; \backslash mathrm\{sinc\}^2\; \backslash left(\; \backslash frac\{\backslash omega\}\{2\backslash pi\; a\}\; \backslash right)$	\frac{1}{ >a	Dual of rule 12.
14	$e^\{-\backslash alpha\; t^2\}\backslash ,$	$\backslash frac\{1\}\{\backslash sqrt\{2\; \backslash alpha\}\}\backslash cdot\; e^\{-\backslash frac\{\backslash omega^2\}\{4\; \backslash alpha\}\}$	$\backslash sqrt\{\backslash frac\{\backslash pi\}\{\backslash alpha\}\}\backslash cdot\; e^\{-\backslash frac\{(\backslash pi\; f)^2\}\{\backslash alpha\}\}$	Shows that the Gaussian function $\backslash exp(-\backslash alpha\; t^2)$ is its own Fourier transform. For this to be integrable we must have $\backslash mathrm\{Re\}(\backslash alpha)>0$.
	e^{i a t^2} = \left. e^{-\alpha t^2}\right >_{\alpha = -i a} \,	$\backslash frac\{1\}\{\backslash sqrt\{2\; a\}\}\; \backslash cdot\; e^\{-i\; \backslash left(\backslash frac\{\backslash omega^2\}\{4\; a\}\; -\backslash frac\{\backslash pi\}\{4\}\backslash right)\}$	$\backslash sqrt\{\backslash frac\{\backslash pi\}\{a\}\}\; \backslash cdot\; e^\{-i\; \backslash left(\backslash frac\{\backslash pi^2\; f^2\}\{a\}\; -\backslash frac\{\backslash pi\}\{4\}\backslash right)\}$	common in optics
	$\backslash cos\; (\; a\; t^2\; )\; \backslash ,$	$\backslash frac\{1\}\{\backslash sqrt\{2\; a\}\}\; \backslash cos\; \backslash left(\; \backslash frac\{\backslash omega^2\}\{4\; a\}\; -\; \backslash frac\{\backslash pi\}\{4\}\; \backslash right)$	$\backslash sqrt\{\backslash frac\{\backslash pi\}\{a\}\}\; \backslash cos\; \backslash left(\; \backslash frac\{\backslash pi^2\; f^2\}\{a\}\; -\; \backslash frac\{\backslash pi\}\{4\}\; \backslash right)$
	$\backslash sin\; (\; a\; t^2\; )\; \backslash ,$	$\backslash frac\{-1\}\{\backslash sqrt\{2\; a\}\}\; \backslash sin\; \backslash left(\; \backslash frac\{\backslash omega^2\}\{4\; a\}\; -\; \backslash frac\{\backslash pi\}\{4\}\; \backslash right)$	$-\; \backslash sqrt\{\backslash frac\{\backslash pi\}\{a\}\}\; \backslash sin\; \backslash left(\; \backslash frac\{\backslash pi^2\; f^2\}\{a\}\; -\; \backslash frac\{\backslash pi\}\{4\}\; \backslash right)$
	e^{-a >t	$\backslash sqrt\{\backslash frac\{2\}\{\backslash pi\}\}\; \backslash cdot\; \backslash frac\{a\}\{a^2\; +\; \backslash omega^2\}$	$\backslash frac\{2\; a\}\{a^2\; +\; 4\; \backslash pi^2\; f^2\}$	a>0
	\frac{1}{\sqrt{ >t	\frac{1}{\sqrt{ >\omega	\frac{1}{\sqrt{ >f	the transform is the function itself
	$J\_0\; (t)\backslash ,$	$\backslash sqrt\{\backslash frac\{2\}\{\backslash pi\}\}\; \backslash cdot\; \backslash frac\{\backslash mathrm\{rect\}\; \backslash left(\; \backslash frac\{\backslash omega\}\{2\}\; \backslash right)\}\{\backslash sqrt\{1\; -\; \backslash omega^2\}\}$	$\backslash frac\{2\backslash cdot\; \backslash mathrm\{rect\}\; (\backslash pi\; f)\}\{\backslash sqrt\{1\; -\; 4\; \backslash pi^2\; f^2\}\}$	J0(t) is the Bessel function of first kind of order 0, rect is the rectangular function
	$J\_n\; (t)\; \backslash ,$	$\backslash sqrt\{\backslash frac\{2\}\{\backslash pi\}\}\; \backslash frac\{\; (-i)^n\; T\_n\; (\backslash omega)\; \backslash mathrm\{rect\}\; \backslash left(\; \backslash frac\{\backslash omega\}\{2\}\; \backslash right)\}\{\backslash sqrt\{1\; -\; \backslash omega^2\}\}$	$\backslash frac\{2\; (-i)^n\; T\_n\; (2\; \backslash pi\; f)\; \backslash mathrm\{rect\}\; (\backslash pi\; f)\}\{\backslash sqrt\{1\; -\; 4\; \backslash pi^2\; f^2\}\}$	it's the generalization of the previous transform; Tn (t) is the Chebyshev polynomial of the first kind.
	$\backslash frac\{J\_n\; (t)\}\{t\}\; \backslash ,$	$\backslash sqrt\{\backslash frac\{2\}\{\backslash pi\}\}\; \backslash frac\{i\}\{n\}\; (-i)^n\; \backslash cdot\; U\_\{n-1\}\; (\backslash omega)\backslash ,$	$\backslash frac\{2\; i\}\{n\}\; (-i)^n\; \backslash cdot\; U\_\{n-1\}\; (2\; \backslash pi\; f)\backslash ,$	Un (t) is the Chebyshev polynomial of the second kind

Distributions

	Signal	Fourier transform unitary, angular frequency	Fourier transform unitary, ordinary frequency	Remarks
	$g(t)\backslash !\backslash equiv\backslash !$ $\backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\backslash !\backslash !G(\backslash omega)\; e^\{i\; \backslash omega\; t\}\; d\; \backslash omega\; \backslash ,$	$G(\backslash omega)\backslash !\backslash equiv\backslash !$ $\backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\backslash !\backslash !g(t)\; e^\{-i\; \backslash omega\; t\}\; dt\; \backslash ,$	$G(f)\backslash !\backslash equiv$ $\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\backslash !\backslash !g(t)\; e^\{-i\; 2\backslash pi\; f\; t\}\; dt\; \backslash ,$
15	$1\backslash ,$	$\backslash sqrt\{2\backslash pi\}\backslash cdot\; \backslash delta(\backslash omega)\backslash ,$	$\backslash delta(f)\backslash ,$	$\backslash delta(\backslash omega)$ denotes the Dirac delta distribution. This rule shows why the Dirac delta is important: it shows up as the Fourier transform of a constant function.
16	$\backslash delta(t)\backslash ,$	$\backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\backslash ,$	$1\backslash ,$	Dual of rule 15.
17	$e^\{i\; a\; t\}\backslash ,$	$\backslash sqrt\{2\; \backslash pi\}\backslash cdot\; \backslash delta(\backslash omega\; -\; a)\backslash ,$	$\backslash delta(f\; -\; \backslash frac\{a\}\{2\backslash pi\})\backslash ,$	This follows from and 3 and 15.
18	$\backslash cos\; (a\; t)\backslash ,$	$\backslash sqrt\{2\; \backslash pi\}\; \backslash frac\{\backslash delta(\backslash omega\backslash !-\backslash !a)\backslash !+\backslash !\backslash delta(\backslash omega\backslash !+\backslash !a)\}\{2\}\backslash ,$	$\backslash frac\{\backslash delta(f\backslash !-\backslash !\backslash begin\{matrix\}\backslash frac\{a\}\{2\backslash pi\}\backslash end\{matrix\})\backslash !+\backslash !\backslash delta(f\backslash !+\backslash !\backslash begin\{matrix\}\backslash frac\{a\}\{2\backslash pi\}\backslash end\{matrix\})\}\{2\}\backslash ,$	Follows from rules 1 and 17 using Euler's formula: $\backslash cos(a\; t)\; =\; (e^\{i\; a\; t\}\; +\; e^\{-i\; a\; t\})/2.$
19	$\backslash sin(\; at)\backslash ,$	$\backslash sqrt\{2\; \backslash pi\}\backslash frac\{\backslash delta(\backslash omega\backslash !-\backslash !a)\backslash !-\backslash !\backslash delta(\backslash omega\backslash !+\backslash !a)\}\{2i\}\backslash ,$	$\backslash frac\{\backslash delta(f\backslash !-\backslash !\backslash begin\{matrix\}\backslash frac\{a\}\{2\backslash pi\}\backslash end\{matrix\})\backslash !-\backslash !\backslash delta(f\backslash !+\backslash !\backslash begin\{matrix\}\backslash frac\{a\}\{2\backslash pi\}\backslash end\{matrix\})\}\{2i\}\backslash ,$	Also from 1 and 17.
20	$t^n\backslash ,$	$i^n\; \backslash sqrt\{2\backslash pi\}\; \backslash delta^\{(n)\}\; (\backslash omega)\backslash ,$	$\backslash left(\backslash frac\{i\}\{2\backslash pi\}\backslash right)^n\; \backslash delta^\{(n)\}\; (f)\backslash ,$	Here, $n$ is a natural number. $\backslash delta^n(\backslash omega)$ is the $n$-th distribution derivative of the Dirac delta. This rule follows from rules 7 and 15. Combining this rule with 1, we can transform all polynomials.
21	$\backslash frac\{1\}\{t\}\backslash ,$	$-i\backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}\backslash sgn(\backslash omega)\backslash ,$	$-i\backslash pi\backslash cdot\; \backslash sgn(f)\backslash ,$	Here $\backslash sgn(\backslash omega)$ is the sign function; note that this is consistent with rules 7 and 15.
22	$\backslash frac\{1\}\{t^n\}\backslash ,$	$-i\; \backslash begin\{matrix\}\; \backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}\backslash cdot\; \backslash frac\{(-i\backslash omega)^\{n-1\}\}\{(n-1)!\}\backslash end\{matrix\}\; \backslash sgn(\backslash omega)\backslash ,$	$-i\backslash pi\; \backslash begin\{matrix\}\; \backslash frac\{(-i\; 2\backslash pi\; f)^\{n-1\}\}\{(n-1)!\}\backslash end\{matrix\}\; \backslash sgn(f)\backslash ,$	Generalization of rule 21.
23	$\backslash sgn(t)\backslash ,$	$\backslash sqrt\{\backslash frac\{2\}\{\backslash pi\}\}\backslash cdot\; \backslash frac\{1\}\{i\backslash \; \backslash omega\; \}\backslash ,$	$\backslash frac\{1\}\{i\backslash pi\; f\}\backslash ,$	The dual of rule 21.
24	$u(t)\; \backslash ,$	$\backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}\; \backslash left(\; \backslash frac\{1\}\{i\; \backslash pi\; \backslash omega\}\; +\; \backslash delta(\backslash omega)\backslash right)\backslash ,$	$\backslash frac\{1\}\{2\}\backslash left(\backslash frac\{1\}\{i\; \backslash pi\; f\}\; +\; \backslash delta(f)\backslash right)\backslash ,$	Here $u(t)$ is the Heaviside unit step function; this follows from rules 1 and 21.
	$e^\{-\; a\; t\}\; u(t)\; \backslash ,$	$\backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\; (a\; +\; i\; \backslash omega)\}$	$\backslash frac\{1\}\{a\; +\; i\; 2\; \backslash pi\; f\}$	$u(t)$ is the Heaviside unit step function and $a\; >\; 0$.
25	$\backslash sum\_\{n=-\backslash infty\}^\{\backslash infty\}\; \backslash delta\; (t\; -\; n\; T)\; \backslash ,$	$\backslash begin\{matrix\}\; \backslash frac\{\backslash sqrt\{2\backslash pi\; \}\}\{T\}\backslash end\{matrix\}\; \backslash sum\_\{k=-\backslash infty\}^\{\backslash infty\}\; \backslash delta\; \backslash left(\; \backslash omega\; -k\; \backslash begin\{matrix\}\; \backslash frac\{2\backslash pi\; \}\{T\}\backslash end\{matrix\}\; \backslash right)\backslash ,$	$\backslash frac\{1\}\{T\}\; \backslash sum\_\{k=-\backslash infty\}^\{\backslash infty\}\; \backslash delta\; \backslash left(\; f\; -\backslash frac\{k\; \}\{T\}\backslash right)\; \backslash ,$	The Dirac comb — helpful for explaining or understanding the transition from continuous to discrete time.

See also

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• Fourier transform
• Fourier series
• Laplace transform
• Discrete Fourier transform

References

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• Fourier Transforms from eFunda - includes tables
• Dym & McKean, Fourier Series and Integrals. (For readers with a background in mathematical analysis.)
• K. Yosida, Functional Analysis, Springer-Verlag, 1968. ISBN 3540586547
• L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, 1976. (Somewhat terse.)
• A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4

External links

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• Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.

Fourier analysis | Integral transforms | Unitary operators

Kontinuierliche Fourier-Transformation | התמרת פורייה | การแปลงฟูริเยร์ต่อเนื่อง | Biến đổi Fourier liên tục | 连续傅里叶变换