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Encyclopedia > Fourier transform
Fourier transforms
Continuous Fourier transform
Fourier series
Discrete Fourier transform
Discrete-time Fourier transform
Related transforms
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In mathematics, the 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. A useful analogy is the relationship between a series of pure notes (the frequency components) and a musical chord (the function itself). In mathematical physics, the Fourier transform of a signal $x(t),$ 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 and linear canonical transform for generalizations.) The Fourier series is a mathematical tool used for analyzing an arbitrary periodic function by decomposing it into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... In mathematics, the discrete Fourier transform (DFT), sometimes called the finite Fourier transform, is a Fourier transform widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal, solve partial differential equations, and to perform other operations such as convolutions. ... A discrete-time Fourier transform (or DTFT) is a Fourier transform of a function of an integer (discrete) time variable n with an unbounded domain. ... This is a list of linear transformations of functions related to the Fourier transform. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... Partial plot of a function f. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ... The Fourier series is a mathematical tool used for analyzing an arbitrary periodic function by decomposing it into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... The fractional Fourier transform (FRFT) is a linear transformation generalizing the continuous Fourier transform, and it can be thought of as the Fourier transform to the n-th power where n need not be an integer &#8212; thus, it can transform a function to an intermediate domain between time and... Paraxial optical systems implemented entirely with thin lenses and propagation through free space and/or graded index (GRIN) media, are Quadratic Phase Systems (QPS). ...

Suppose $x,$ is a complex-valued Lebesgue integrable function. The Fourier transform to the frequency domain, $omega,$, is given by the function: In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... The integral can be interpreted as the area under a curve. ...

$X(omega) = frac{1}{sqrt{2pi}} int_{-infty}^infty x(t) e^{- iomega t},dt$,   for every real number $omega ,$.

When the independent variable t represents time (with SI unit of seconds), the transform variable ω represents angular frequency (in radians per second). In mathematics, the real numbers may be described informally in several different ways. ... Cover of brochure The International System of Units. ... Look up second in Wiktionary, the free dictionary. ... It has been suggested that this article or section be merged into Angular velocity. ... In mathematics and physics, the radian is a unit of angle measure. ...

Other notations for this same function are:  $hat{x}(omega),$  and  $mathcal{F}{x}(omega),$.  The function is complex-valued in general.   ($i,$ represents the imaginary unit.) In mathematics, the imaginary unit (sometimes also represented by the Latin or the Greek iota) allows the real number system to be extended to the complex number system . ...

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

$x(t) = frac{1}{sqrt{2pi}} int_{-infty}^{infty} X(omega) e^{ iomega t},domega$,   for every real number $t ,$.

The interpretation of $X(omega),$ is aided by expressing it in polar coordinate form, $X(omega) = A(omega )cdot e^{i phi (omega )} ,$, where: This article describes some of the common coordinate systems that appear in elementary mathematics. ...

$A(omega ) = |X(omega)| ,$ the amplitude
$phi (omega ) = angle X(omega) ,$ the phase

Then the inverse transform can be written: Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation, that is, magnitude of the maximum disturbance in the medium during one wave cycle. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

$x(t) = frac{1}{sqrt{2pi}} int_{-infty}^{infty} A(omega) e^{ i(omega t +phi (omega ))},domega$

which is a recombination of all the frequency components of $x(t),$.  Each component is a complex sinusoid of the form $e^{ iomega t},$ whose amplitude is proportional to $A(omega),$ and whose initial phase angle (at t = 0) is $phi (omega ),$. Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation, that is, magnitude of the maximum disturbance in the medium during one wave cycle. ... The phase angle of a point on a periodic wave is the distance between the point and a specified reference point, expressed using an angular measure. ...

## Normalization factors and alternative forms

The factors $1oversqrt{2pi}$ before each integral ensure that there is no net change in amplitude when one transforms from one domain to the other and back. The actual requirement is that their product be  $1 over 2 pi$.  When they are chosen to be equal, the transform is referred to as unitary. A common non-unitary convention is shown here: In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ...

$X(omega) = int_{-infty}^infty x(t) e^{- iomega t},dt$
$x(t) = frac{1}{2pi} int_{-infty}^{infty} X(omega) e^{ iomega t},domega$

As a rule of thumb, mathematicians generally prefer the unitary transform (for symmetry reasons), and physicists use either convention depending on the application.

The non-unitary form is preferred by some engineers as a special case of the bilateral Laplace transform. And the substitution:   $omega = 2pi f,$, where $f,$ is ordinary frequency (hertz), results in another unitary transform that is popular in the field of signal processing and communications systems: In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ... Sine waves of various frequencies; the bottom waves have higher frequencies than those above. ... The hertz (symbol: Hz) is the SI unit of frequency. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ...

$X(f) = int_{-infty}^infty x(t) e^{-i 2pi f t},dt$
$x(t) = int_{-infty}^infty X(f) e^{i 2pi f t},df$

We note that $X(f),$ and $X(omega),$ represent different, but related, functions, as shown in the table below.

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.

angular frequency $omega ,$ (rad/s) $X_1(omega) stackrel{mathrm{def}}{=} frac{1}{sqrt{2 pi}} int_{-infty}^{infty} x(t) e^{-i omega t}, dt = frac{1}{sqrt{2 pi}} X_2(omega) = frac{1}{sqrt{2 pi}} X_3(frac{omega}{2 pi}),$ $x(t) = frac{1}{sqrt{2 pi}} int_{-infty}^{infty} X_1(omega) e^{i omega t}, d omega$ $X_2(omega) stackrel{mathrm{def}}{=} int_{-infty}^{infty} x(t) e^{-i omega t} dt = sqrt{2 pi} X_1(omega) = X_3(frac{omega}{2 pi}),$ $x(t) = frac{1}{2 pi} int_{-infty}^{infty} X_2(omega) e^{i omega t} d omega$ $X_3(f) stackrel{mathrm{def}}{=} int_{-infty}^{infty} x(t) e^{-i 2 pi f t} dt = sqrt{2 pi} X_1(2 pi f) = X_2(2 pi f),$ $x(t) = int_{-infty}^{infty} X_3(f) e^{i 2 pi f t}, df$

## Generalization

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: In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

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

and the inverse is given by:

$x(t) = sqrt{frac{|b|}{(2 pi)^{1+a}}} int_{-infty}^{+infty} X(omega) e^{i b omega t} , domega$

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) = (0,2π) which is often used in signal processing applications. In this case, the angular frequency ω 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). Look up second in Wiktionary, the free dictionary. ... The hertz (symbol: Hz) is the SI unit of frequency. ...

## Properties

In this section, all the results are derived for the following definition (normalization) of the Fourier transform:

$F(omega) = mathcal{F}{f}(omega) = int_{-infty}^infty f(t) e^{- iomega t},dt$

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

### Completeness

We define the Fourier transform on the set of compactly-supported complex-valued functions of $mathbb{R}$ and then extend it by continuity to the Hilbert space of square-integrable functions with the usual inner-product. Then $mathcal{F}:L^2(mathbb{R})rightarrow L^2(mathbb{R})$ is a unitary operator. That is. $mathcal{F}^*=mathcal{F}^{-1}$ and the transform preserves inner-products (see Parseval's theorem, also described below). Note that, $mathcal{F}^*$ refers to adjoint of the Fourier Transform operator. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In functional analysis, it is often convenient to define something on a normed vector space by defining it on a dense set and extending it to the whole space. ... In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ... In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ... In mathematics, Parsevals theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. ... In mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator. ...

Moreover we can check that,

$mathcal{F}^2 = mathcal{J},quad mathcal{F}^3 = mathcal{F}^*=mathcal{F}^{-1}, quad mbox{and} quad mathcal{F}^4 = mathcal{I}quad$

where $mathcal{J}$ is the Time-Reversal operator defined as,

$||mathcal{J}{f}(t) - f(-t)||_2 =0$

and $mathcal{I}$ is the Identity operator defined as,

$||mathcal{I}{f}(t) - f(t)||_2 =0$

### Extensions

The Fourier transform can also be extended to the space integrable functions defined on $mathbb{R}^n$

$mathcal{F}:L^1(mathbb{R}^n)rightarrow C(mathbb{R}^n).$

where,

$L^1(mathbb{R}^n) = {f: , mathbb{R}^n to mathbb{C} ;big|; int_{mathbb{R}^n} |f(x)| dx < infty}.$

and $C(mathbb{R}^n)$ is the space of continuous functions on $mathbb{R}^n$. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

In this case the definition usually appears as

$mathcal{F}{f}(w) stackrel{mathrm{def}}{=} int_{R^n} f(x)e^{-iomegacdot x},dx.$

where $omegain mathbb{R}^n$ and $omega cdot x$ is the inner product of the two vectors ω and x. In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

One may now use this to define the continuous Fourier transform for compactly supported smooth functions, which are dense in $L^2(mathbb{R}^n).$ The Plancherel theorem then allows us to extend the definition of the Fourier transform to functions on $L^2(mathbb{R}^n)$ (even those not compactly supported) by continuity arguments. All the properties and formulas listed on this page apply to the Fourier transform so defined. In mathematics, the Plancherel theorem is a result in harmonic analysis, first proved by Michel Plancherel. ...

Unfortunately, further extensions become more technical. One may use the Hausdorff-Young inequality to define the Fourier transform for $fin L^p(mathbb{R}^n)$ for $1leq pleq 2$. The Fourier transform of functions in Lp for the range $2 requires the study of distributions, since the Fourier transform of some functions in these spaces is no longer a function, but rather a distribution. In mathematics, the Riesz-Thorin theorem, often referred to as the Riesz-Thorin Interpolation Theorem or the Riesz-Thorin Convexity Theorem is a result about interpolation of operators. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ...

### The Plancherel theorem and Parseval's theorem

It should be noted that depending on the author either of these theorems might be referred to as the Plancherel theorem or as Parseval's theorem.

If f(t) and g(t) are square-integrable and F(ω) and G(ω) are their Fourier transforms, then we have Parseval's theorem: In mathematics, Parsevals theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. ...

$int_{mathbb{R}^n} f(t) bar{g}(t) , dt = int_{mathbb{R}^n} F(omega) bar{G}(omega) , domega,$

where the bar denotes complex conjugation. Therefore, the Fourier transformation yields an isometric automorphism of the Hilbert space $L^2(mathbb{R}^n)$. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ...

The Plancherel theorem, a special case of Parseval's theorem, states that In mathematics, the Plancherel theorem is a result in harmonic analysis, first proved by Michel Plancherel. ... In mathematics, Parsevals theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. ...

$int_{mathbb{R}^n} left| f(t) right|^2 dt = int_{mathbb{R}^n} left| F(omega) right|^2 domega.$

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. In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ...

### Localization property

As a rule of thumb: the more concentrated f(t) is, the more spread out is F(ω). In particular, if we "squeeze" a function in t, it spreads out in ω and vice-versa; and we cannot arbitrarily concentrate both the function and its Fourier transform.

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 Gaussian function Gaussian curves parametrised by expected value and variance (see normal distribution) A Gaussian function (named after Carl Friedrich Gauss) is a function of the form: for some real constants a > 0, b, and c. ...

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

This function is related to Gaussian distributions, and in fact, is an eigenfunction of the Fourier transform operators. Probability density function of Gaussian distribution (bell curve). ... In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ...

The trade-off between the compaction of a function and its Fourier transform can be formalized. Suppose f(t) and F(ω) are a Fourier transform pair. Without loss of generality, we assume that f(t) is normalized:

$int_{-infty}^infty f(t)bar{f}(t),dt=1.$

It follows from Parseval's theorem that F(ω) is also normalized. Define the expected value of a function A(t) as: In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

$langle Arangle stackrel{mathrm{def}}{=} int_{-infty}^infty A(t)f(t)bar{f}(t),dt$

and the expectation value of a function B(ω) as:

$langle Brangle stackrel{mathrm{def}}{=} int_{-infty}^infty B(omega)F(omega)bar{F}(omega),domega$

Also define the variance of A(t) as: In probability theory and statistics, the variance of a random variable (or equivalently, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

$Delta^2 A stackrel{mathrm{def}}{=} langle (A-langle Arangle) ^2rangle$

and similarly define the variance of B(ω). Then it can be shown that

$Delta t Delta omega ge frac{1}{2}.$

The equality is achieved for the Gaussian function listed above, which shows that the gaussian function is maximally concentrated in "time-frequency".

The most famous practical application of this property is found in quantum mechanics. The momentum and position wave functions are Fourier transform pairs to within a factor of $h over 2 pi$ and are normalized to unity. The above expression then becomes a statement of the Heisenberg uncertainty principle. Fig. ... In quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. ...

The Fourier transform also translates between smoothness and decay: if f(t) is several times differentiable, then F(ω) decays rapidly towards zero for $s to plusmn infin$.

### 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 $mathbb{R}^n$ can also be translated into algebraic equations. In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, a derivative is the rate of change of a quantity (e. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...

### Convolution theorem

Main article: Convolution theorem

The Fourier transform translates between convolution and multiplication of functions. If f(t) and h(t) are integrable functions with Fourier transforms F(ω) and H(ω) respectively, and if the convolution of f and h exists and is absolutely integrable, then the Fourier transform of the convolution is given by the product of the Fourier transforms F(ω)H(ω) (possibly multiplied by a constant factor depending on the Fourier normalization convention). In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the point-wise product of Fourier transforms. ... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ...

In the current normalization convention, this means that if

$g(t) = {f*h}(t) = int_{-infty}^infty f(s)h(t - s),ds$

where * denotes the convolution operation; then

$G(omega) = sqrt{2pi}cdot F(omega)H(omega).,$

The above formulas hold true for functions defined on both one- and multi-dimension real space. In linear time invariant (LTI) system theory, it is common to interpret h(t) as the impulse response of an LTI system with input f(t) and output g(t), since substituting the unit impulse for f(t) yields g(t) = h(t). In this case, H(ω) represents the frequency response of the system. In electrical engineering, specifically in signal processing and control theory, LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. ... In the language of mathematics, the impulse response of a linear transformation is the image of Diracs delta function under the transformation. ... The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0, the value zero elsewhere. ... Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ...

Conversely, if f(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(ω) and Q(ω), again with a constant scaling factor.

In the current normalization convention, this means that if

$f(t) = p(t) q(t),$

then

$F(omega) = frac{1}{sqrt{2pi}} bigg( P(omega) * Q(omega) bigg) = frac{1}{sqrt{2pi}} int_{-infty}^infty P(alpha)Q(omega - alpha),dalpha.$

### Cross-correlation theorem

In an analogous manner, it can be shown that if g(t) is the cross-correlation of f(t) and h(t): In statistics, the term cross-correlation is sometimes used to refer to the covariance cov(X, Y) between two random vectors X and Y, in order to distinguish that concept from the covariance of a random vector X, which is understood to be the matrix of covariances between the scalar...

$g(t)=(fstar h)(t) = int_{-infty}^infty bar{f}(s),h(t+s),ds$

then the Fourier transform of g(t) is:

$G(omega) = sqrt{2pi},bar{F}(omega),H(omega)$

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

### Tempered distributions

The most general and useful 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 $1/sqrt{2pi}$. Distributions can be differentiated and the above mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions. In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0, the value zero elsewhere. ...

## Table of important Fourier transforms

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
$g(t)! stackrel{mathrm{def}}{=} !$

$frac{1}{sqrt{2 pi}} int_{-infty}^{infty}!!G(omega) e^{i omega t} d omega ,$
$G(omega)! stackrel{mathrm{def}}{=} !$

$frac{1}{sqrt{2 pi}} int_{-infty}^{infty}!!g(t) e^{-i omega t} dt ,$
$G(f)! stackrel{mathrm{def}}{=}$

$int_{-infty}^{infty}!!g(t) e^{-i 2pi f t} dt ,$
1 $acdot g(t) + bcdot h(t),$ $acdot G(omega) + bcdot H(omega),$ $acdot G(f) + bcdot H(f),$ Linearity
2 $g(t - a),$ $e^{- i a omega} G(omega),$ $e^{- i 2pi a f} G(f),$ Shift in time domain
3 $e^{ iat} g(t),$ $G(omega - a),$ $G left(f - frac{a}{2pi}right),$ Shift in frequency domain, dual of 2
4 $g(a t),$ $frac{1}{|a|} G left( frac{omega}{a} right),$ $frac{1}{|a|} G left( frac{f}{a} right),$ If $|a|,$ is large, then $g(a t),$ is concentrated around 0 and $frac{1}{|a|}G left( frac{omega}{a} right),$ spreads out and flattens. It is interesting to consider the limit of this as | a | tends to infinity - the delta function
5 $G(t),$ $g(-omega),$ $g(-f),$ Duality property of the Fourier transform. Results from swapping "dummy" variables of $t ,$ and $omega ,$.
6 $frac{d^n g(t)}{dt^n},$ $(iomega)^n G(omega),$ $(i 2pi f)^n G(f),$ Generalized derivative property of the Fourier transform
7 $t^n g(t),$ $i^n frac{d^n G(omega)}{domega^n},$ $left (frac{i}{2pi}right)^n frac{d^n G(f)}{df^n},$ This is the dual of 6
8 $(g * h)(t),$ $sqrt{2pi} G(omega) H(omega),$ $G(f) H(f),$ $g * h,$ denotes the convolution of $g,$ and $h,$ — this rule is the convolution theorem
9 $g(t) h(t),$ $(G * H)(omega) over sqrt{2pi},$ $(G * H)(f),$ This is the dual of 8

In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the point-wise product of Fourier transforms. ...

### Square-integrable functions

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks
$g(t)! stackrel{mathrm{def}}{=} !$

$frac{1}{sqrt{2 pi}} int_{-infty}^{infty}!!G(omega) e^{i omega t} mathrm{d} omega ,$
$G(omega)! stackrel{mathrm{def}}{=} !$

$frac{1}{sqrt{2 pi}} int_{-infty}^{infty}!!g(t) e^{-i omega t} mathrm{d}t ,$
$G(f)! stackrel{mathrm{def}}{=}$

$int_{-infty}^{infty}!!g(t) e^{-i 2pi f t} mathrm{d}t ,$
10 $mathrm{rect}(a t) ,$ $frac{1}{sqrt{2 pi a^2}}cdot mathrm{sinc}left(frac{omega}{2pi a}right)$ $frac{1}{|a|}cdot mathrm{sinc}left(frac{f}{a}right)$ The rectangular pulse and the normalized sinc function
11 $mathrm{sinc}(a t),$ $frac{1}{sqrt{2pi a^2}}cdot mathrm{rect}left(frac{omega}{2 pi a}right)$ $frac{1}{|a|}cdot mathrm{rect}left(frac{f}{a} right),$ 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 $mathrm{sinc}^2 (a t) ,$ $frac{1}{sqrt{2pi a^2}}cdot mathrm{tri} left( frac{omega}{2pi a} right)$ $frac{1}{|a|}cdot mathrm{tri} left( frac{f}{a} right)$ tri is the triangular function
13 $mathrm{tri} (a t) ,$ $frac{1}{sqrt{2pi a^2}} cdot mathrm{sinc}^2 left( frac{omega}{2pi a} right)$ $frac{1}{|a|}cdot mathrm{sinc}^2 left( frac{f}{a} right) ,$ Dual of rule 12.
14 $e^{-alpha t^2},$ $frac{1}{sqrt{2 alpha}}cdot e^{-frac{omega^2}{4 alpha}}$ $sqrt{frac{pi}{alpha}}cdot e^{-frac{(pi f)^2}{alpha}}$ Shows that the Gaussian function exp( − αt2) is its own Fourier transform. For this to be integrable we must have Re(α) > 0.
15 $e^{iat^2} = left. e^{-alpha t^2}right|_{alpha = -i a} ,$ $frac{1}{sqrt{2 a}} cdot e^{-i left(frac{omega^2}{4 a} -frac{pi}{4}right)}$ $sqrt{frac{pi}{a}} cdot e^{-i left(frac{pi^2 f^2}{a} -frac{pi}{4}right)}$ common in optics
16 $cos ( a t^2 ) ,$ $frac{1}{sqrt{2 a}} cos left( frac{omega^2}{4 a} - frac{pi}{4} right)$ $sqrt{frac{pi}{a}} cos left( frac{pi^2 f^2}{a} - frac{pi}{4} right)$
17 $sin ( a t^2 ) ,$ $frac{-1}{sqrt{2 a}} sin left( frac{omega^2}{4 a} - frac{pi}{4} right)$ $- sqrt{frac{pi}{a}} sin left( frac{pi^2 f^2}{a} - frac{pi}{4} right)$
18 $mathrm{e}^{-a|t|} ,$ $sqrt{frac{2}{pi}} cdot frac{a}{a^2 + omega^2}$ $frac{2 a}{a^2 + 4 pi^2 f^2}$ a>0
19 $frac{1}{sqrt{|t|}} ,$ $frac{1}{sqrt{|omega|}}$ $frac{1}{sqrt{|f|}}$ the transform is the function itself
20 $J_0 (t),$ $sqrt{frac{2}{pi}} cdot frac{mathrm{rect} left( frac{omega}{2} right)}{sqrt{1 - omega^2}}$ $frac{2cdot mathrm{rect} (pi f)}{sqrt{1 - 4 pi^2 f^2}}$ J0(t) is the Bessel function of first kind of order 0
21 $J_n (t) ,$ $sqrt{frac{2}{pi}} frac{ (-i)^n T_n (omega) mathrm{rect} left( frac{omega}{2} right)}{sqrt{1 - omega^2}}$ $frac{2 (-i)^n T_n (2 pi f) mathrm{rect} (pi f)}{sqrt{1 - 4 pi^2 f^2}}$ it's the generalization of the previous transform; Tn (t) is the Chebyshev polynomial of the first kind.
22 $frac{J_n (t)}{t} ,$ $sqrt{frac{2}{pi}} frac{i}{n} (-i)^n cdot U_{n-1} (omega),$

$cdot sqrt{1 - omega^2} mathrm{rect} left( frac{omega}{2} right)$ The rectangular function (also known as the rectangle function or the normalized boxcar function) is defined as or in terms of the Heaviside step function The rectangular function is normalized: The Fourier transform of the rectangular function is where sinc is the sinc function. ... The sinc function sinc(x) from x = âˆ’8Ï€ to 8Ï€. In mathematics, the sinc function (for sinus cardinalis), also known as the interpolation function, filtering function or the first spherical Bessel function , is the product of a sine function and a monotonically decreasing function. ... The rectangular function (also known as the rectangle function or the normalized boxcar function) is defined as or in terms of the Heaviside step function The rectangular function is normalized: The Fourier transform of the rectangular function is where sinc is the sinc function. ... A low-pass filter is a filter that passes low frequencies well, but attenuates (or reduces) frequencies higher than the cutoff frequency. ... The sinc function sinc(x) from x = âˆ’8Ï€ to 8Ï€. In mathematics, the sinc function (for sinus cardinalis), also known as the interpolation function, filtering function or the first spherical Bessel function , is the product of a sine function and a monotonically decreasing function. ... An acausal system is a system that depends on both the past and the future. ... The triangular function (also known as the triangle function, hat function, or tent function) is defined as: or, equivalently, as the convolution of two rectangular functions: The function is useful in signal processing and communication systems engineering as a representation of an idealized signal, and as a prototype or kernel... Gaussian curves parametrised by expected value and variance (see normal distribution) A Gaussian function (named after Carl Friedrich Gauss) is a function of the form: for some real constants a > 0, b, and c. ... Table of Opticks, 1728 Cyclopaedia Optics ( appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. ... In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number Î± (the order). ... In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivres formula and which are easily defined recursively, like Fibonacci or Lucas numbers. ...

$frac{2 mathrm{i}}{n} (-i)^n cdot U_{n-1} (2 pi f),$

$cdot sqrt{1 - 4 pi^2 f^2} mathrm{rect} ( pi f )$

Un (t) is the Chebyshev polynomial of the second kind

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivres formula and which are easily defined recursively, like Fibonacci or Lucas numbers. ...

### Distributions

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks
$g(t)! stackrel{mathrm{def}}{=} !$

$frac{1}{sqrt{2 pi}} int_{-infty}^{infty}!!G(omega) e^{i omega t} d omega ,$
$G(omega)! stackrel{mathrm{def}}{=} !$

$frac{1}{sqrt{2 pi}} int_{-infty}^{infty}!!g(t) e^{-i omega t} dt ,$
$G(f)! stackrel{mathrm{def}}{=}$

$int_{-infty}^{infty}!!g(t) e^{-i 2pi f t} dt ,$
23 $1,$ $sqrt{2pi}cdot delta(omega),$ $delta(f),$ δ(ω) 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.
24 $delta(t),$ $frac{1}{sqrt{2pi}},$ $1,$ Dual of rule 23.
25 $e^{i a t},$ $sqrt{2 pi}cdot delta(omega - a),$ $delta(f - frac{a}{2pi}),$ This follows from and 3 and 24.
26 $cos (a t),$ $sqrt{2 pi} frac{delta(omega!-!a)!+!delta(omega!+!a)}{2},$ $frac{delta(f!-!begin{matrix}frac{a}{2pi}end{matrix})!+!delta(f!+!begin{matrix}frac{a}{2pi}end{matrix})}{2},$ Follows from rules 1 and 25 using Euler's formula: cos(at) = (eiat + e iat) / 2.
27 $sin( at),$ $sqrt{2 pi}frac{delta(omega!-!a)!-!delta(omega!+!a)}{2i},$ $frac{delta(f!-!begin{matrix}frac{a}{2pi}end{matrix})!-!delta(f!+!begin{matrix}frac{a}{2pi}end{matrix})}{2i},$ Also from 1 and 25.
28 $t^n,$ $i^n sqrt{2pi} delta^{(n)} (omega),$ $left(frac{i}{2pi}right)^n delta^{(n)} (f),$ Here, n is a natural number. δn(ω) is the n-th distribution derivative of the Dirac delta. This rule follows from rules 7 and 24. Combining this rule with 1, we can transform all polynomials.
29 $frac{1}{t},$ $-isqrt{frac{pi}{2}}sgn(omega),$ $-ipicdot sgn(f),$ Here sgn(ω) is the sign function; note that this is consistent with rules 7 and 24.
30 $frac{1}{t^n},$ $-i begin{matrix} sqrt{frac{pi}{2}}cdot frac{(-iomega)^{n-1}}{(n-1)!}end{matrix} sgn(omega),$ $-ipi begin{matrix} frac{(-i 2pi f)^{n-1}}{(n-1)!}end{matrix} sgn(f),$ Generalization of rule 29.
31 $sgn(t),$ $sqrt{frac{2}{pi}}cdot frac{1}{i omega },$ $frac{1}{ipi f},$ The dual of rule 29.
32 $u(t) ,$ $sqrt{frac{pi}{2}} left( frac{1}{i pi omega} + delta(omega)right),$ $frac{1}{2}left(frac{1}{i pi f} + delta(f)right),$ Here u(t) is the Heaviside unit step function; this follows from rules 1 and 31.
33 $e^{- a t} u(t) ,$ $frac{1}{sqrt{2 pi} (a + i omega)}$ $frac{1}{a + i 2 pi f}$ u(t) is the Heaviside unit step function and a > 0.
34 $sum_{n=-infty}^{infty} delta (t - n T) ,$ $begin{matrix} frac{sqrt{2pi }}{T}end{matrix} sum_{k=-infty}^{infty} delta left( omega -k begin{matrix} frac{2pi }{T}end{matrix} right),$ $frac{1}{T} sum_{k=-infty}^{infty} delta left( f -frac{k }{T}right) ,$ The Dirac comb — helpful for explaining or understanding the transition from continuous to discrete time.

The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function &#948;(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ... This article is about the Eulers formula in complex analysis. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... Signum function In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. ... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of... In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions for some given period T. Some authors, notably Bracewell, refer to it as the Shah function (probably because its graph resembles the shape of the cyrillic letter sha Ð¨). From the orthogonality of the Fourier series...

## Fourier transform properties

Notation: $f(t) iff F(omega)$ denotes that $f(t),$ and $F(omega),$ are a Fourier transform pair.

Conjugation
$overline{f(t)} iff overline{F(-omega)}$
Scaling
$f(at) iff frac{1}{|a|}Fbiggl(frac{omega}{a}biggr), qquad a in mathbb{R}, a ne 0$
Time reversal
$f(-t) iff F(-omega)$
Time shift
$f(t-t_0) iff e^{-iomega t_0}F(omega)$
Modulation (multiplication by complex exponential)
$f(t)cdot e^{iomega_{0}t} iff F(omega-omega_{0})qquad omega_{0} in mathbb{R},$
Multiplication by sin ω0t
$f(t)sin omega_{0}t iff frac{i}{2}[F(omega+omega_{0})-F(omega-omega_{0})],$
Multiplication by cos ω0t
$f(t)cos omega_{0}t iff frac{1}{2}[F(omega+omega_{0})+F(omega-omega_{0})],$
Integration
$int_{-infty}^{t} f(u), du iff frac{1}{jomega}F(omega)+pi F(0)delta(omega),$
Parseval's theorem
$int_{mathbb{R}} f(t)cdot overline{g(t)}, dt = int_{mathbb{R}} F(omega)cdot overline{G(omega)}, domega ,$

The Fourier series is a mathematical tool used for analyzing an arbitrary periodic function by decomposing it into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... In mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ... In mathematics, the discrete Fourier transform (DFT), sometimes called the finite Fourier transform, is a Fourier transform widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal, solve partial differential equations, and to perform other operations such as convolutions. ... The fractional Fourier transform (FRFT) is a linear transformation generalizing the continuous Fourier transform, and it can be thought of as the Fourier transform to the n-th power where n need not be an integer &#8212; thus, it can transform a function to an intermediate domain between time and... Paraxial optical systems implemented entirely with thin lenses and propagation through free space and/or graded index (GRIN) media, are Quadratic Phase Systems (QPS). ...

## References

• 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 3-540-58654-7
• 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

This page may meet Wikipediaâ€™s criteria for speedy deletion. ... Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ...

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