integral transforms and delta function

8/12/2019 Integral Transforms and Delta Function

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Contents

Articles

Fourier transform 1

Convolution 23

Convolution theorem 35

Laplace transform 37

Dirac delta function 54

References

Article Sources and Contributors 76

Image Sources, Licenses and Contributors 77

Article Licenses

License 78


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Fourier transform 1

Fourier transform

Fourier transforms

Continuous Fourier transform

Fourier series

Discrete-time Fourier transform

Discrete Fourier transform

Fourier analysis

Related transforms

The Fourier transform (English pronunciation:/frie/), named after Joseph Fourier, is a mathematical transformation

employed to transform signals between time (or spatial) domain and frequency domain, which has many applications

in physics and engineering. It is reversible, being able to transform from either domain to the other. The term itself

refers to both the transform operation and to the function it produces.In the case of a periodic function over time (for example, a continuous but not necessarily sinusoidal musical sound),

the Fourier transform can be simplified to the calculation of a discrete set of complex amplitudes, called Fourier

series coefficients. They represent the frequency spectrum of the original time-domain signal. Also, when a

time-domain function is sampled to facilitate storage or computer-processing, it is still possible to recreate a version

of the original Fourier transform according to the Poisson summation formula, also known as discrete-time Fourier

transform. See also Fourier analysis and List of Fourier-related transforms.

Definition

There are several common conventions for defining the Fourier transform of an integrable function(Kaiser 1994, p. 29), (Rahman 2011, p. 11). This article will use the following definition:

, for any real number .

When the independent variable x represents time (with SI unit of seconds), the transform variable represents

frequency (in hertz). Under suitable conditions, is determined by via the inverse transform:

, for any real numberx.

The statement that can be reconstructed from is known as the Fourier inversion theorem, and was first

introduced in Fourier'sAnalytical Theory of Heat(Fourier 1822, p. 525), (Fourier & Freeman 1878, p. 408), although

what would be considered a proof by modern standards was not given until much later (Titchmarsh 1948, p. 1). The

functions and often are referred to as aFourier integral pair orFourier transform pair(Rahman 2011, p. 10).

For other common conventions and notations, including using the angular frequency instead of the frequency ,

see Other conventions and Other notations below. The Fourier transform on Euclidean space is treated separately, in

which the variablex often represents position and momentum.
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Fourier transform 2

Introduction

The Fourier transform relates the function's time

domain, shown in red, to the function's frequency

domain, shown in blue. The component

frequencies, spread across the frequency

spectrum, are represented as peaks in the

frequency domain.

The motivation for the Fourier transform comes from the study of

Fourier series. In the study of Fourier series, complicated but periodic

functions are written as the sum of simple waves mathematically

represented by sines and cosines. The Fourier transform is an extension

of the Fourier series that results when the period of the represented

function is lengthened and allowed to approach infinity (Taneja 2008,

p. 192).

Due to the properties of sine and cosine, it is possible to recover the

amplitude of each wave in a Fourier series using an integral. In many

cases it is desirable to use Euler's formula, which states that e2i =

cos(2) + i sin(2), to write Fourier series in terms of the basic

waves e2i. This has the advantage of simplifying many of the

formulas involved, and provides a formulation for Fourier series that

more closely resembles the definition followed in this article.Re-writing sines and cosines as complex exponentials makes it

necessary for the Fourier coefficients to be complex valued. The usual interpretation of this complex number is that

it gives both the amplitude (or size) of the wave present in the function and the phase (or the initial angle) of the

wave. These complex exponentials sometimes contain negative "frequencies". If is measured in seconds, then the

waves e2i and e2i both complete one cycle per second, but they represent different frequencies in the Fourier

transform. Hence, frequency no longer measures the number of cycles per unit time, but is still closely related.

There is a close connection between the definition of Fourier series and the Fourier transform for functions f which

are zero outside of an interval. For such a function, we can calculate its Fourier series on any interval that includes

the points wheref is not identically zero. The Fourier transform is also defined for such a function. As we increase

the length of the interval on which we calculate the Fourier series, then the Fourier series coefficients begin to look

like the Fourier transform and the sum of the Fourier series off begins to look like the inverse Fourier transform. To

explain this more precisely, suppose that T is large enough so that the interval [T/2, T/2] contains the interval on

whichf is not identically zero. Then the n-th series coefficient cn

is given by:

Comparing this to the definition of the Fourier transform, it follows that since f(x) is zero

outside [T/2,T/2]. Thus the Fourier coefficients are just the values of the Fourier transform sampled on a grid of

width 1/T, multiplied by the grid width 1/T.

Under appropriate conditions, the sum of the Fourier series of f will equal the function f. In other words, f can bewritten:

where the last sum is simply the first sum rewritten using the definitions n

= n/T, and = (n + 1)/T n/T = 1/T.

This second sum is a Riemann sum, and so by letting T it will converge to the integral for the inverse Fourier

transform given in the definition section. Under suitable conditions this argument may be made precise (Stein &

Shakarchi 2003).

In the study of Fourier series the numbers cn

could be thought of as the "amount" of the wave present in the Fourier

series off. Similarly, as seen above, the Fourier transform can be thought of as a function that measures how muchof each individual frequency is present in our function f, and we can recombine these waves by using an integral (or
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Fourier transform 3

"continuous sum") to reproduce the original function.

Example

The following images provide a visual illustration of how the Fourier transform measures whether a frequency is

present in a particular function. The function depicted f(t) = cos(6t) et2 oscillates at 3 hertz (if t measures

seconds) and tends quickly to 0. (The second factor in this equation is an envelope function that shapes thecontinuous sinusoid into a short pulse. Its general form is a Gaussian function). This function was specially chosen to

have a real Fourier transformwhich can easily be plotted. The first image contains its graph. In order to calculate

we must integrate e2i(3t)f(t). The second image shows the plot of the real and imaginary parts of this

function. The real part of the integrand is almost always positive, because when f(t) is negative, the real part of

e2i(3t) is negative as well. Because they oscillate at the same rate, when f(t) is positive, so is the real part of

e2i(3t). The result is that when you integrate the real part of the integrand you get a relatively large number (in this

case 0.5). On the other hand, when you try to measure a frequency that is not present, as in the case when we look at

, the integrand oscillates enough so that the integral is very small. The general situation may be a bit more

complicated than this, but this in spirit is how the Fourier transform measures how much of an individual frequency

is present in a functionf(t).

Original function showing

oscillation 3 hertz.

Real and imaginary parts of

integrand for Fourier transform

at 3 hertz

Real and imaginary parts of

integrand for Fourier transform

at 5 hertz

Fourier transform with 3 and 5

hertz labeled.

Properties of the Fourier transform

Here we assumef(x), g(x) and h(x) are integrable functions, are Lebesgue-measurable on the real line, and satisfy:

We denote the Fourier transforms of these functions by , and respectively.

Basic properties

The Fourier transform has the following basic properties: (Pinsky 2002).

Linearity

For any complex numbers a and b, if h(x) = af(x) + bg(x), then

Translation

For any real numberx0, if then

Modulation

For any real number 0

if then

Scaling
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Fourier transform 5

Uniform continuity and the RiemannLebesgue lemma

The rectangular function is Lebesgue integrable.

The sinc function, which is the Fourier transform

of the rectangular function, is bounded and

continuous, but not Lebesgue integrable.

The Fourier transform may be defined in some cases for non-integrable

functions, but the Fourier transforms of integrable functions have

several strong properties.

The Fourier transform, , of any integrable function f is uniformly

continuous and (Katznelson 1976). By the

RiemannLebesgue lemma (Stein & Weiss 1971),

However, need not be integrable. For example, the Fourier

transform of the rectangular function, which is integrable, is the sinc

function, which is not Lebesgue integrable, because its improper

integrals behave analogously to the alternating harmonic series, in

converging to a sum without being absolutely convergent.

It is not generally possible to write the inverse transform as a Lebesgue

integral. However, when both f and are integrable, the inverse

equality

holds almost everywhere. That is, the Fourier transform is injective on

L1(R). (But iff is continuous, then equality holds for everyx.)

Plancherel theorem and Parseval's theorem

Let f(x) and g(x) be integrable, and let and be their Fourier transforms. If f(x) and g(x) are also

square-integrable, then we have Parseval's theorem (Rudin 1987, p. 187):

where the bar denotes complex conjugation.

The Plancherel theorem, which is equivalent to Parseval's theorem, states (Rudin 1987, p. 186):

The Plancherel theorem makes it possible to extend the Fourier transform, by a continuity argument, to a unitary

operator on L2(R). On L1(R) L2(R), this extension agrees with original Fourier transform defined on L1(R), thus

enlarging the domain of the Fourier transform to L1(R) + L2(R) (and consequently to Lp(R) for 1 p 2). ThePlancherel theorem has the interpretation in the sciences that the Fourier transform preserves the energy of the

original quantity. Depending on the author either of these theorems might be referred to as the Plancherel theorem or

as Parseval's theorem.

See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.
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Fourier transform 6

Poisson summation formula

The Poisson summation formula (PSF) is an equation that relates the Fourier series coefficients of the periodic

summation of a function to values of the function's continuous Fourier transform. It has a variety of useful forms that

are derived from the basic one by application of the Fourier transform's scaling and time-shifting properties. The

frequency-domain dual of the standard PSF is also called discrete-time Fourier transform, which leads directly to:

a popular, graphical, frequency-domain representation of the phenomenon of aliasing, and a proof of the Nyquist-Shannon sampling theorem.

Convolution theorem

The Fourier transform translates between convolution and multiplication of functions. Iff(x) and g(x) are integrable

functions with Fourier transforms and respectively, then the Fourier transform of the convolution is

given by the product of the Fourier transforms and (under other conventions for the definition of the

Fourier transform a constant factor may appear).This means that if:

where denotes the convolution operation, then:

In linear time invariant (LTI) system theory, it is common to interpret g(x) as the impulse response of an LTI system

with input f(x) and output h(x), since substituting the unit impulse for f(x) yields h(x) = g(x). In this case,

represents the frequency response of the system.

Conversely, if f(x) can be decomposed as the product of two square integrable functions p(x) and q(x), then the

Fourier transform off(x) is given by the convolution of the respective Fourier transforms and .

Cross-correlation theorem

In an analogous manner, it can be shown that if h(x) is the cross-correlation off(x) and g(x):

then the Fourier transform of h(x) is:

As a special case, the autocorrelation of functionf(x) is:

for which
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Fourier transform 7

Eigenfunctions

One important choice of an orthonormal basis forL2(R) is given by the Hermite functions

where Hen

(x) are the "probabilist's" Hermite polynomials, defined by

Under this convention for the Fourier transform, we have that

.

In other words, the Hermite functions form a complete orthonormal system of eigenfunctions for the Fourier

transform on L2(R) (Pinsky 2002). However, this choice of eigenfunctions is not unique. There are only four

different eigenvalues of the Fourier transform (1 and i) and any linear combination of eigenfunctions with the

same eigenvalue gives another eigenfunction. As a consequence of this, it is possible to decompose L2(R) as a direct

sum of four spacesH0

,H1

,H2

, andH3

where the Fourier transform acts onHek

simply by multiplication by ik.

Since the complete set of Hermite functions provides a resolution of the identity, the Fourier transform can be

represented by such a sum of terms weighted by the above eigenvalues, and these sums can be explicitly summed.

This approach to define the Fourier transform was first done by Norbert Wiener (Duoandikoetxea 2001). Among

other properties, Hermite functions decrease exponentially fast in both frequency and time domains, and they are

thus used to define a generalization of the Fourier transform, namely the fractional Fourier transform used in

time-frequency analysis (Boashash 2003). In physics, this transform was introduced by Edward Condon (Condon

1937).

Fourier transform on Euclidean space

The Fourier transform can be in any arbitrary number of dimensions n. As with the one-dimensional case, there aremany conventions. For an integrable functionf(x), this article takes the definition:

where x and are n-dimensional vectors, and x is the dot product of the vectors. The dot product is sometimes

written as .

All of the basic properties listed above hold for the n-dimensional Fourier transform, as do Plancherel's and

Parseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and the

RiemannLebesgue lemma holds. (Stein & Weiss 1971)

Uncertainty principle

Generally speaking, the more concentrated f(x) is, the more spread out its Fourier transform must be. In

particular, the scaling property of the Fourier transform may be seen as saying: if we "squeeze" a function in x, its

Fourier transform "stretches out" in . It is not possible to arbitrarily concentrate both a function and its Fourier

transform.

The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an

uncertainty principle by viewing a function and its Fourier transform as conjugate variables with respect to the

symplectic form on the timefrequency domain: from the point of view of the linear canonical transformation, the

Fourier transform is rotation by 90 in the timefrequency domain, and preserves the symplectic form.

Suppose f(x) is an integrable and square-integrable function. Without loss of generality, assume that f(x) is

normalized:
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Fourier transform 8

It follows from the Plancherel theorem that is also normalized.

The spread aroundx= 0 may be measured by the dispersion about zero(Pinsky 2002, p. 131) defined by

In probability terms, this is the second moment of |f(x)|2 about zero.

The Uncertainty principle states that, if f(x) is absolutely continuous and the functions xf(x) and f(x) are square

integrable, then

(Pinsky 2002).

The equality is attained only in the case (hence ) where > 0 is

arbitrary and C1

is such that f is L2normalized (Pinsky 2002). In other words, where f is a (normalized) Gaussian

function with variance 2, centered at zero, and its Fourier transform is a Gaussian function with variance 2.

In fact, this inequality implies that:

for anyx0,

0 R (Stein & Shakarchi 2003, p. 158).

In quantum mechanics, the momentum and position wave functions are Fourier transform pairs, to within a factor of

Planck's constant. With this constant properly taken into account, the inequality above becomes the statement of the

Heisenberg uncertainty principle (Stein & Shakarchi 2003, p. 158).

A stronger uncertainty principle is the Hirschman uncertainty principle which is expressed as:

whereH(p) is the differential entropy of the probability density functionp(x):

where the logarithms may be in any base which is consistent. The equality is attained for a Gaussian, as in the

previous case.

Spherical harmonics

Let the set of homogeneous harmonic polynomials of degree k on Rn be denoted by Ak. The set A

kconsists of the

solid spherical harmonics of degree k. The solid spherical harmonics play a similar role in higher dimensions to the

Hermite polynomials in dimension one. Specifically, if f(x) = e|x|2P(x) for some P(x) in Ak, then

. Let the set Hk

be the closure inL2(Rn) of linear combinations of functions of the form f(|x|)P(x)

where P(x) is in Ak. The space L2(Rn) is then a direct sum of the spaces H

kand the Fourier transform maps each

space Hk

to itself and is possible to characterize the action of the Fourier transform on each space Hk

(Stein & Weiss

1971). Letf(x) =f0(|x|)P(x) (withP(x) in A

k), then where

Here J(n+ 2k2)/2

denotes the Bessel function of the first kind with order (n+ 2k2)/2. When k= 0 this gives a

useful formula for the Fourier transform of a radial function (Grafakos 2004). Note that this is essentially the Hankel

transform. Moreover, there is a simple recursion relating the cases n+2 and n (Grafakos & Teschl 2013) allowing to

compute, e.g., the three-dimensional Fourier transform of a radial function from the one-dimensional one.
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Fourier transform 10

where the limit is taken in the L2 sense. Many of the properties of the Fourier transform in L1 carry over toL2, by a

suitable limiting argument.

Furthermore : L2(Rn) L2(Rn) is a unitary operator (Stein & Weiss 1971, Thm. 2.3). For an operator to be

unitary it is sufficient to show that it is bijective and preserves the inner product, so in this case these follow from the

Fourier inversion theorem combined with the fact that for anyf,gL2(Rn) we have

In particular, the image ofL2(Rn) is itself under the Fourier transform.

On otherLp

The definition of the Fourier transform can be extended to functions in Lp(Rn) for 1 p 2 by decomposing such

functions into a fat tail part in L2 plus a fat body part in L1. In each of these spaces, the Fourier transform of a

function in Lp(Rn) is in Lq(Rn), where is the Hlder conjugate of p. by the HausdorffYoung

inequality. However, except for p = 2, the image is not easily characterized. Further extensions become more

technical. The Fourier transform of functions in Lp for the range 2 < p < requires the study of distributions

(Katznelson 1976). In fact, it can be shown that there are functions in Lp

withp> 2 so that the Fourier transform isnot defined as a function (Stein & Weiss 1971).

Tempered distributions

One might consider enlarging the domain of the Fourier transform fromL1+L2 by considering generalized functions,

or distributions. A distribution on Rn is a continuous linear functional on the space Cc(Rn) of compactly supported

smooth functions, equipped with a suitable topology. The strategy is then to consider the action of the Fourier

transform on Cc(Rn) and pass to distributions by duality. The obstruction to do this is that the Fourier transform does

not map Cc(Rn) to C

c(Rn). In fact the Fourier transform of an element in C

c(Rn) can not vanish on an open set; see

the above discussion on the uncertainty principle. The right space here is the slightly larger space of Schwartz

functions. The Fourier transform is an automorphism on the Schwartz space, as a topological vector space, and thusinduces an automorphism on its dual, the space of tempered distributions(Stein & Weiss 1971). The tempered

distribution include all the integrable functions mentioned above, as well as well-behaved functions of polynomial

growth and distributions of compact support.

For the definition of the Fourier transform of a tempered distribution, let f and g be integrable functions, and let

and be their Fourier transforms respectively. Then the Fourier transform obeys the following multiplication

formula (Stein & Weiss 1971),

Every integrable functionf defines (induces) a distribution Tf by the relation

for all Schwartz functions .

So it makes sense to define Fourier transform of Tf

by

for all Schwartz functions . Extending this to all tempered distributions T gives the general definition of the Fourier

transform.

Distributions can be differentiated and the above mentioned compatibility of the Fourier transform with

differentiation and convolution remains true for tempered distributions.
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Generalizations

FourierStieltjes transform

The Fourier transform of a finite Borel measure on Rnis given by (Pinsky 2002, p. 256):

This transform continues to enjoy many of the properties of the Fourier transform of integrable functions. One

notable difference is that the RiemannLebesgue lemma fails for measures (Katznelson 1976). In the case that d =

f(x)dx, then the formula above reduces to the usual definition for the Fourier transform of f. In the case that is the

probability distribution associated to a random variable X, the Fourier-Stieltjes transform is closely related to the

characteristic function, but the typical conventions in probability theory take eix instead of e2ix (Pinsky 2002).

In the case when the distribution has a probability density function this definition reduces to the Fourier transform

applied to the probability density function, again with a different choice of constants.

The Fourier transform may be used to give a characterization of measures. Bochner's theorem characterizes which

functions may arise as the FourierStieltjes transform of a positive measure on the circle (Katznelson 1976).

Furthermore, the Dirac delta function is not a function but it is a finite Borel measure. Its Fourier transform is a

constant function (whose specific value depends upon the form of the Fourier transform used).

Locally compact abelian groups

The Fourier transform may be generalized to any locally compact abelian group. A locally compact abelian group is

an abelian group which is at the same time a locally compact Hausdorff topological space so that the group operation

is continuous. If G is a locally compact abelian group, it has a translation invariant measure , called Haar measure.

For a locally compact abelian group G, the set of irreducible, i.e. one-dimensional, unitary representations are called

its characters. With its natural group structure and the topology of pointwise convergence, the set of characters is

itself a locally compact abelian group, called the Pontryagin dual of G. For a function f in L1(G), its Fouriertransform is defined by (Katznelson 1976):

The Riemann-Lebesgue lemma holds in this case; is a function vanishing at infinity on .

Gelfand transform

The Fourier transform is also a special case of Gelfand transform. In this particular context, it is closely related to the

Pontryagin duality map defined above.

Given an abelian locally compact Hausdorff topological group G, as before we consider spaceL1(G), defined using aHaar measure. With convolution as multiplication, L1(G) is an abelian Banach algebra. It also has an involution *

given by

Taking the completion with respect to the largest possibly C*-norm gives its enveloping C*-algebra, called the group

C*-algebra C*(G) of G. (Any C*-norm onL1(G) is bounded by theL1 norm, therefore their supremum exists.)

Given any abelian C*-algebra A, the Gelfand transform gives an isomorphism between A and C0(A^), whereA^ is

the multiplicative linear functionals, i.e. one-dimensional representations, on A with the weak-* topology. The map

is simply given by
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It turns out that the multiplicative linear functionals of C*(G), after suitable identification, are exactly the characters

of G, and the Gelfand transform, when restricted to the dense subsetL1(G) is the Fourier-Pontryagin transform.

Non-abelian groups

The Fourier transform can also be defined for functions on a non-abelian group, provided that the group is compact.

Removing the assumption that the underlying group is abelian, irreducible unitary representations need not alwaysbe one-dimensional. This means the Fourier transform on a non-abelian group takes values as Hilbert space operators

(Hewitt & Ross 1970, Chapter 8). The Fourier transform on compact groups is a major tool in representation theory

(Knapp 2001) and non-commutative harmonic analysis.

Let G be a compact Hausdorff topological group. Let denote the collection of all isomorphism classes of

finite-dimensional irreducible unitary representations, along with a definite choice of representation U() on the

Hilbert spaceH

of finite dimension d

for each . If is a finite Borel measure on G, then the FourierStieltjes

transform of is the operator onH

defined by

where is the complex-conjugate representation of U() acting onH. If is absolutely continuous with respect

to the left-invariant probability measure on G, represented as

for somef L1(), one identifies the Fourier transform off with the FourierStieltjes transform of .

The mapping defines an isomorphism between the Banach spaceM(G) of finite Borel measures (see rca

space) and a closed subspace of the Banach space C

() consisting of all sequences E = (E) indexed by of

(bounded) linear operatorsE:H

H

for which the norm

is finite. The "convolution theorem" asserts that, furthermore, this isomorphism of Banach spaces is in fact anisometric isomorphism of C* algebras into a subspace of C

(). Multiplication onM(G) is given by convolution of

measures and the involution * defined by

and C

() has a natural C*-algebra structure as Hilbert space operators.

The Peter-Weyl theorem holds, and a version of the Fourier inversion formula (Plancherel's theorem) follows: if f

L2(G), then

where the summation is understood as convergent in theL2

sense.The generalization of the Fourier transform to the noncommutative situation has also in part contributed to the

development of noncommutative geometry.[citation needed] In this context, a categorical generalization of the Fourier

transform to noncommutative groups is Tannaka-Krein duality, which replaces the group of characters with the

category of representations. However, this loses the connection with harmonic functions.
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Alternatives

In signal processing terms, a function (of time) is a representation of a signal with perfect time resolution, but no

frequency information, while the Fourier transform has perfect frequency resolution, but no time information: the

magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by

phase (argument of the Fourier transform at a point), and standing waves are not localized in time a sine wave

continues out to infinity, without decaying. This limits the usefulness of the Fourier transform for analyzing signalsthat are localized in time, notably transients, or any signal of finite extent.

As alternatives to the Fourier transform, in time-frequency analysis, one uses time-frequency transforms or

time-frequency distributions to represent signals in a form that has some time information and some frequency

information by the uncertainty principle, there is a trade-off between these. These can be generalizations of the

Fourier transform, such as the short-time Fourier transform or fractional Fourier transform, or other functions to

represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous)

Fourier transform being the continuous wavelet transform. (Boashash 2003).

Applications

Some problems, such as certain differential equations, become easier to solve when the

Fourier transform is applied. In that case the solution to the original problem is recovered

using the inverse Fourier transform.

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: iff(x) is a differentiable function

with Fourier transform , then the Fourier transform of its derivative is given by . This can be used

to transform differential equations into algebraic equations. 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 partial

differential equations with domain Rn can also be translated into algebraic equations.

Fourier transform spectroscopy

The Fourier transform is also used in nuclear magnetic resonance (NMR) and in other kinds of spectroscopy, e.g.

infrared (FTIR). In NMR an exponentially shaped free induction decay (FID) signal is acquired in the time domain

and Fourier-transformed to a Lorentzian line-shape in the frequency domain. The Fourier transform is also used in

magnetic resonance imaging (MRI) and mass spectrometry.

Quantum mechanics and signal processing

In quantum mechanics, Fourier transforms of solutions to the Schrdinger equation are known as momentum space

(or k space) wave functions. They display the amplitudes for momenta. Their absolute square is the probabilities of

momenta. This is valid also for classical waves treated in signal processing, such as in swept frequency radar where

data is taken in frequency domain and transformed to time domain, yielding range. The absolute square is then the

power.
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Other notations

Other common notations for include:

Denoting the Fourier transform by a capital letter corresponding to the letter of function being transformed (such as

f(x) and F()) is especially common in the sciences and engineering. In electronics, the omega () is often used

instead of due to its interpretation as angular frequency, sometimes it is written as F(j), wherej is the imaginary

unit, to indicate its relationship with the Laplace transform, and sometimes it is written informally asF(2f) in order

to use ordinary frequency.

The interpretation of the complex function may be aided by expressing it in polar coordinate form

in terms of the two real functionsA() and () where:

is the amplitude and

is the phase (see arg function).

Then the inverse transform can be written:

which is a recombination of all the frequency components of f(x). Each component is a complex sinusoid of the

form e2ix whose amplitude isA() and whose initial phase angle (atx= 0) is ().

The Fourier transform may be thought of as a mapping on function spaces. This mapping is here denoted and

is used to denote the Fourier transform of the function f. This mapping is linear, which means that can

also be seen as a linear transformation on the function space and implies that the standard notation in linear algebraof applying a linear transformation to a vector (here the function f) can be used to write instead of .

Since the result of applying the Fourier transform is again a function, we can be interested in the value of this

function evaluated at the value for its variable, and this is denoted either as or as . Notice that

in the former case, it is implicitly understood that is applied first tof and then the resulting function is evaluated

at , not the other way around.In mathematics and various applied sciences it is often necessary to distinguish between a functionf and the value of

f when its variable equals x, denotedf(x). This means that a notation like formally can be interpreted as

the Fourier transform of the values off atx. Despite this flaw, the previous notation appears frequently, often when a

particular function or a function of a particular variable is to be transformed.

For example, is sometimes used to express that the Fourier transform of a rectangular

function is a sinc function,

or is used to express the shift property of the Fourier transform.

Notice, that the last example is only correct under the assumption that the transformed function is a function ofx, not

ofx0.
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Other conventions

The Fourier transform can also be written in terms of angular frequency: = 2 whose units are radians per second.

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

Under this convention, the inverse transform becomes:

Unlike the convention followed in this article, when the Fourier transform is defined this way, it is no longer a

unitary transformation onL2(Rn). There is also less symmetry between the formulas for the Fourier transform and its

inverse.

Another convention is to split the factor of (2)n evenly between the Fourier transform and its inverse, which leads

to definitions:

Under this convention, the Fourier transform is again a unitary transformation on L2(Rn). It also restores the

symmetry between the Fourier transform and its inverse.

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.

Summary of popular forms of the Fourier transform

ordinary frequency (hertz) unitary

angular frequency (rad/s) non-unitary

unitary

As discussed above, the characteristic function of a random variable is the same as the FourierStieltjes transform of

its distribution measure, but in this context it is typical to take a different convention for the constants. Typically

characteristic function is defined .

As in the case of the "non-unitary angular frequency" convention above, there is no factor of 2 appearing in either

of the integral, or in the exponential. Unlike any of the conventions appearing above, this convention takes the

opposite sign in the exponential.
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Tables of important Fourier transforms

The following tables record some closed-form Fourier transforms. For functions f(x), g(x) and h(x) denote their

Fourier transforms by , , and respectively. Only the three most common conventions are included. It may

be useful to notice that entry 105 gives a relationship between the Fourier transform of a function and the original

function, which can be seen as relating the Fourier transform and its inverse.

Functional relationships

The Fourier transforms in this table may be found in Erdlyi (1954) or Kammler (2000, appendix).

Function Fourier transform

unitary, ordinary

frequency

Fourier transform

unitary, angular frequency

Fourier transform

non-unitary, angular

frequency

Remarks

Definition

101 Linearity

102 Shift in time domain

103 Shift in frequency domain, dual

of 102

104 Scaling in the time domain. If

is large, then is

concentrated around 0 and

spreads out and

flattens.

105 Duality. Here needs to be

calculated using the same

method as Fourier transform

column. Results from swapping

"dummy" variables of and

or or .

106

107 This is the dual of 106

108 The notation denotes the

convolution of and this

rule is the convolution theorem

109 This is the dual of 108

110 For purely real Hermitian symmetry.

indicates the complex

conjugate.

111 For a purely real

even function

, and are purely real even functions.

112 For a purely real

odd function

, and are purely imaginary odd functions.

113 Complex conjugation,

generalization of 110
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Square-integrable functions

The Fourier transforms in this table may be found in (Campbell & Foster 1948), (Erdlyi 1954), or the appendix of

(Kammler 2000).

Function Fourier transformunitary, ordinary

frequency

Fourier transformunitary, angular frequency

Fourier transformnon-unitary, angular

frequency

Remarks

201 The rectangular pulse and the

normalizedsinc function, here defined

as sinc(x) = sin(x)/(x)

202 Dual of rule 201. The rectangular

function is an ideal low-pass filter, and

the sinc function is the non-causalimpulse response of such a filter. The

sinc function is defined here as sinc(x)

= sin(x)/(x)

203 The function tri(x) is the triangular

function

204 Dual of rule 203.

205 The function u(x) is the Heaviside unit

step function and a>0.

206This shows that, for the unitary Fouriertransforms, the Gaussian function

exp(x2) is its own Fourier transform

for some choice of . For this to be

integrable we must have Re()>0.

207 For a>0. That is, the Fourier transform

of a decaying exponential function is a

Lorentzian function.

208 Hyperbolic secant is its own Fourier

transform

209 is the Hermite's polynomial. If a =

1 then the Gauss-Hermite functions areeigenfunctions of the Fourier transform

operator. For a derivation, see Hermite

polynomial. The formula reduces to

206 for n = 0.
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Distributions

The Fourier transforms in this table may be found in (Erdlyi 1954) or the appendix of (Kammler 2000).


unitary, ordinary frequency

Fourier transform


Fourier transform

non-unitary, angular

frequency

Remarks

301 The distribution ()

denotes the Dirac delta

function.

302 Dual of rule 301.

303 This follows from 103

and 301.

304 This follows from rules101 and 303 using Euler's

formula:

305 This follows from 101

and 303 using

306

307

308 Here, n is a natural

number and is

the n-th distribution

derivative of the Dirac

delta function. This rule

follows from rules 107

and 301. Combining this

rule with 101, we can

transform all

polynomials.

309 Here sgn() is the signfunction. Note that 1/x is

not a distribution. It is

necessary to use the

Cauchy principal value

when testing against

Schwartz functions. This

rule is useful in studying

the Hilbert transform.

310 1/xn is the homogeneous

distribution defined by

the distributional

derivative
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311 This formula is valid for

0 > > 1. For > 0

some singular terms arise

at the origin that can be

found by differentiating

318. If Re > 1, then

is a locally

integrable function, and

so a tempered

distribution. The function

is a

holomorphic function

from the right half-plane

to the space of tempered

distributions. It admits a

unique meromorphic

extension to a tempered

distribution, also denoted

for 2, 4, ...

(See homogeneous

distribution.)

312 The dual of rule 309. This

time the Fourier

transforms need to be

considered as Cauchy

principal value.

313 The function u(x) is the

Heaviside unit step

function; this follows

from rules 101, 301, and

312.

314 This function is known as

the Dirac comb function.

This result can be derived

from 302 and 102,

together with the fact that

as

distributions.

315 The functionJ0(x) is the

zeroth order Bessel

function of first kind.

316 This is a generalization of

315. The functionJn(x) is

the n-th order Bessel

function of first kind. The

function Tn(x) is the

Chebyshev polynomial of

the first kind.

317 is the

EulerMascheroni

constant.
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318 This formula is valid for

1 > > 0. Use

differentiation to derive

formula for higher

exponents. u is the

Heaviside function.

Two-dimensional functions



Fourier transform


Fourier transform

non-unitary, angular frequency

400

401

402

Remarks

To 400: The variables x

, y

, x

, y

, x

and y

are real numbers. The integrals are taken over the entire plane.

To 401: Both functions are Gaussians, which may not have unit volume.

To 402: The function is defined by circ(r)=1 0r1, and is 0 otherwise. This is the Airy distribution, and is

expressed using J1

(the order 1 Bessel function of the first kind). (Stein & Weiss 1971, Thm. IV.3.3)

Formulas for generaln-dimensional functions



Fourier transform


Fourier transform

non-unitary, angular frequency

500

501

502

503

Remarks

To 501: The function[0, 1]

is the indicator function of the interval [0, 1]. The function (x) is the gamma function.

The functionJn/2 +

is a Bessel function of the first kind, with order n/2 + . Taking n = 2 and = 0 produces 402.

(Stein & Weiss 1971, Thm. 4.15)

To 502: See Riesz potential. The formula also holds for all n, n 1, by analytic continuation, but then the

function and its Fourier transforms need to be understood as suitably regularized tempered distributions. See

homogeneous distribution.
http://en.wikipedia.org/w/index.php?title=Homogeneous_distributionhttp://en.wikipedia.org/w/index.php?title=Riesz_potentialhttp://en.wikipedia.org/w/index.php?title=Indicator_functionhttp://en.wikipedia.org/w/index.php?title=Bessel_function


22/79


To 503: This is the formula for a multivariate normal distribution normalized to 1 with a mean of 0. Bold variables

are vectors or matrices. Following the notation of the aforementioned page, and

References

Boashash, B., ed. (2003), Time-Frequency Signal Analysis and Processing: A Comprehensive Reference, Oxford:

Elsevier Science, ISBN 0-08-044335-4 Bochner S., Chandrasekharan K. (1949),Fourier Transforms, Princeton University Press

Bracewell, R. N. (2000), The Fourier Transform and Its Applications (3rd ed.), Boston: McGraw-Hill,

ISBN 0-07-116043-4.

Campbell, George; Foster, Ronald (1948),Fourier Integrals for Practical Applications, New York: D. Van

Nostrand Company, Inc..

Condon, E. U. (1937), "Immersion of the Fourier transform in a continuous group of functional transformations",

Proc. Nat. Acad. Sci. USA23: 158164.

Duoandikoetxea, Javier (2001),Fourier Analysis, American Mathematical Society, ISBN 0-8218-2172-5.

Dym, H; McKean, H (1985),Fourier Series and Integrals, Academic Press, ISBN 978-0-12-226451-1.

Erdlyi, Arthur, ed. (1954), Tables of Integral Transforms1, New Your: McGraw-Hill Fourier, J. B. Joseph (1822), Thorie Analytique de la Chaleur[1], Paris: Chez Firmin Didot, pre et fils

Fourier, J. B. Joseph; Freeman, Alexander, translator (1878), The Analytical Theory of Heat[2], The University

Press

Grafakos, Loukas (2004), Classical and Modern Fourier Analysis, Prentice-Hall, ISBN 0-13-035399-X.

Grafakos, Loukas; Teschl, Gerald (2013), "On Fourier transforms of radial functions and distributions",J. Fourier

Anal. Appl.19: 167-179, doi:10.1007/s00041-012-9242-5 [3].

Hewitt, Edwin; Ross, Kenneth A. (1970),Abstract harmonic analysis. Vol. II: Structure and analysis for compact

groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band

152, Berlin, New York: Springer-Verlag, MR 0262773 [4].

Hrmander, L. (1976),Linear Partial Differential Operators, Volume 1, Springer-Verlag,

ISBN 978-3-540-00662-6.

James, J.F. (2011),A Student's Guide to Fourier Transforms (3rd ed.), New York: Cambridge University Press,

ISBN 978-0-521-17683-5.

Kaiser, Gerald (1994),A Friendly Guide to Wavelets[5], Birkhuser, ISBN 0-8176-3711-7

Kammler, David (2000),A First Course in Fourier Analysis, Prentice Hall, ISBN 0-13-578782-3

Katznelson, Yitzhak (1976),An introduction to Harmonic Analysis, Dover, ISBN 0-486-63331-4

Knapp, Anthony W. (2001),Representation Theory of Semisimple Groups: An Overview Based on Examples[6],

Princeton University Press, ISBN 978-0-691-09089-4

Pinsky, Mark (2002),Introduction to Fourier Analysis and Wavelets[7], Brooks/Cole, ISBN 0-534-37660-6

Polyanin, A. D.; Manzhirov, A. V. (1998),Handbook of Integral Equations, Boca Raton: CRC Press,

ISBN 0-8493-2876-4.

Rudin, Walter (1987),Real and Complex Analysis (Third ed.), Singapore: McGraw Hill, ISBN 0-07-100276-6.

Rahman, Matiur (2011),Applications of Fourier Transforms to Generalized Functions[8], WIT Press,

ISBN 1845645642.

Stein, Elias; Shakarchi, Rami (2003),Fourier Analysis: An introduction[9], Princeton University Press,

ISBN 0-691-11384-X.

Stein, Elias; Weiss, Guido (1971),Introduction to Fourier Analysis on Euclidean Spaces[10], Princeton, N.J.:

Princeton University Press, ISBN 978-0-691-08078-9.

Taneja, HC (2008), "Chapter 18: Fourier integrals and Fourier transforms" [11],Advanced Engineering

Mathematics:, Volume 2, New Delhi, India: I. K. International Pvt Ltd, ISBN 8189866567.
http://en.wikipedia.org/w/index.php?title=Special:BookSources/8189866567http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://books.google.com/books?id=X-RFRHxMzvYC&pg=PA192&dq=%22The+Fourier+integral+can+be+regarded+as+an+extension+of+the+concept+of+Fourier+series%22&hl=en&sa=X&ei=D4rDT_vdCueQiAKF6PWeCA&ved=0CDQQ6AEwAA#v=onepage&q=%22The%20Fourier%20integral%20can%20be%20regarded%20as%20an%20extension%20of%20the%20concept%20of%20Fourier%20series%22&f=falsehttp://en.wikipedia.org/w/index.php?title=Special:BookSources/978-0-691-08078-9http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://books.google.com/books?id=YUCV678MNAIC&dq=editions:xbArf-TFDSEC&source=gbs_navlinks_shttp://en.wikipedia.org/w/index.php?title=Guido_Weisshttp://en.wikipedia.org/w/index.php?title=Elias_Steinhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/0-691-11384-Xhttp://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://books.google.com/books?id=FAOc24bTfGkC&pg=PA158&dq=%22The+mathematical+thrust+of+the+principle%22&hl=en&sa=X&ei=Esa7T5PZIsqriQKluNjPDQ&ved=0CDQQ6AEwAA#v=onepage&q=%22The%20mathematical%20thrust%20of%20the%20principle%22&f=falsehttp://en.wikipedia.org/w/index.php?title=Special:BookSources/1845645642http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://books.google.com/books?id=k_rdcKaUdr4C&pg=PA10http://en.wikipedia.org/w/index.php?title=Special:BookSources/0-07-100276-6http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/0-8493-2876-4http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/0-534-37660-6http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://books.google.com/books?id=tlLE4KUkk1gC&pg=PA256&dq=%22The+Fourier+transform+of+the+measure%22&hl=en&sa=X&ei=w8e7T43XJsiPiAKZztnRDQ&ved=0CEUQ6AEwAg#v=onepage&q=%22The%20Fourier%20transform%20of%20the%20measure%22&f=falsehttp://en.wikipedia.org/w/index.php?title=Special:BookSources/978-0-691-09089-4http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Princeton_University_Presshttp://books.google.com/?id=QCcW1h835pwChttp://en.wikipedia.org/w/index.php?title=Special:BookSources/0-486-63331-4http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/0-13-578782-3http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/0-8176-3711-7http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://books.google.com/books?id=rfRnrhJwoloC&pg=PA29&dq=%22becomes+the+Fourier+%28integral%29+transform%22&hl=en&sa=X&ei=osO7T7eFOqqliQK3goXoDQ&ved=0CDQQ6AEwAA#v=onepage&q=%22becomes%20the%20Fourier%20%28integral%29%20transform%22&f=falsehttp://en.wikipedia.org/w/index.php?title=Special:BookSources/978-0-521-17683-5http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/978-3-540-00662-6http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Lars_H%C3%B6rmanderhttp://www.ams.org/mathscinet-getitem?mr=0262773http://en.wikipedia.org/w/index.php?title=Mathematical_Reviewshttp://en.wikipedia.org/w/index.php?title=Springer-Verlaghttp://dx.doi.org/10.1007%2Fs00041-012-9242-5http://en.wikipedia.org/w/index.php?title=Digital_object_identifierhttp://en.wikipedia.org/w/index.php?title=Gerald_Teschlhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/0-13-035399-Xhttp://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://books.google.com/books?id=-N8EAAAAYAAJ&pg=PA408&dq=%22that+is+to+say,+that+we+have+the+equation%22&hl=en&sa=X&ei=F667T-u5I4WeiALEwpHXDQ&ved=0CDgQ6AEwAA#v=onepage&q=%22that%20is%20to%20say%2C%20that%20we%20have%20the%20equation%22&f=falsehttp://books.google.com/books?id=TDQJAAAAIAAJ&pg=PA525&dq=%22c%27est-%C3%A0-dire+qu%27on+a+l%27%C3%A9quation%22&hl=en&sa=X&ei=SrC7T9yKBorYiALVnc2oDg&sqi=2&ved=0CEAQ6AEwAg#v=onepage&q=%22c%27est-%C3%A0-dire%20qu%27on%20a%20l%27%C3%A9quation%22&f=falsehttp://en.wikipedia.org/w/index.php?title=Joseph_Fourierhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/978-0-12-226451-1http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Harry_Dymhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/0-8218-2172-5http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Edward_Condonhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/0-07-116043-4http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=K._S._Chandrasekharanhttp://en.wikipedia.org/w/index.php?title=Salomon_Bochnerhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/0-08-044335-4http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Multivariate_normal_distribution


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Titchmarsh, E (1948),Introduction to the theory of Fourier integrals (2nd ed.), Oxford University: Clarendon

Press (published 1986), ISBN 978-0-8284-0324-5.

Wilson, R. G. (1995),Fourier Series and Optical Transform Techniques in Contemporary Optics, New York:

Wiley, ISBN 0-471-30357-7.

Yosida, K. (1968),Functional Analysis, Springer-Verlag, ISBN 3-540-58654-7.

External links

The Discrete Fourier Transformation (DFT): Definition and numerical examples [12] A Matlab tutorial

The Fourier Transform Tutorial Site [13] (thefouriertransform.com)

Fourier Series Applet [14] (Tip: drag magnitude or phase dots up or down to change the wave form).

Stephan Bernsee's FFTlab [15] (Java Applet)

Stanford Video Course on the Fourier Transform [16]

Hazewinkel, Michiel, ed. (2001), "Fourier transform" [17],Encyclopedia of Mathematics, Springer,

ISBN 978-1-55608-010-4

Weisstein, Eric W., "Fourier Transform [18]",MathWorld.

The DFT Pied: Mastering The Fourier Transform in One Day [19] at The DSP Dimension An Interactive Flash Tutorial for the Fourier Transform [20]

Java Library for DFT [21]

References

[1] http://books. google.com/books?id=TDQJAAAAIAAJ&pg=PA525&dq=%22c%27est-%C3%A0-dire+qu%27on+ a+

l%27%C3%A9quation%22&hl=en&sa=X&ei=SrC7T9yKBorYiALVnc2oDg&sqi=2&ved=0CEAQ6AEwAg#v=onepage&

q=%22c%27est-%C3%A0-dire%20qu%27on%20a%20l%27%C3%A9quation%22&f=false

[2] http://books. google.com/books?id=-N8EAAAAYAAJ&pg=PA408&dq=%22that+is+to+say,+that+we+have+the+equation%22&

hl=en&sa=X&ei=F667T-u5I4WeiALEwpHXDQ&ved=0CDgQ6AEwAA#v=onepage&

q=%22that%20is%20to%20say%2C%20that%20we%20have%20the%20equation%22&

f=false[3] http://dx.doi. org/10.1007%2Fs00041-012-9242-5

[4] http://www.ams.org/mathscinet-getitem?mr=0262773

[5] http://books. google.com/books?id=rfRnrhJwoloC&pg=PA29&dq=%22becomes+the+Fourier+%28integral%29+transform%22&

hl=en&sa=X&ei=osO7T7eFOqqliQK3goXoDQ&ved=0CDQQ6AEwAA#v=onepage&

q=%22becomes%20the%20Fourier%20%28integral%29%20transform%22&f=false

[6] http://books. google.com/?id=QCcW1h835pwC

[7] http://books. google.com/books?id=tlLE4KUkk1gC&pg=PA256&dq=%22The+Fourier+transform+of+the+measure%22&hl=en&

sa=X&ei=w8e7T43XJsiPiAKZztnRDQ&ved=0CEUQ6AEwAg#v=onepage&

q=%22The%20Fourier%20transform%20of%20the%20measure%22&f=false

[8] http://books. google.com/books?id=k_rdcKaUdr4C&pg=PA10

[9] http://books. google.com/books?id=FAOc24bTfGkC&pg=PA158&dq=%22The+mathematical+thrust+of+the+principle%22&hl=en&

sa=X&ei=Esa7T5PZIsqriQKluNjPDQ&ved=0CDQQ6AEwAA#v=onepage&

q=%22The%20mathematical%20thrust%20of%20the%20principle%22&f=false

[10] http://books. google.com/books?id=YUCV678MNAIC&dq=editions:xbArf-TFDSEC&source=gbs_navlinks_s

[11] http://books. google.com/books?id=X-RFRHxMzvYC&pg=PA192&dq=%22The+Fourier+integral+can+be+regarded+as+an+

extension+of+the+concept+of+Fourier+series%22&hl=en&sa=X&ei=D4rDT_vdCueQiAKF6PWeCA&

ved=0CDQQ6AEwAA#v=onepage&

q=%22The%20Fourier%20integral%20can%20be%20regarded%20as%20an%20extension%20of%20the%20concept%20of%20Fourier%20series%22&

f=false

[12] http://www.nbtwiki.net/doku. php?id=tutorial:the_discrete_fourier_transformation_dft

[13] http://www.thefouriertransform.com

[14] http://www.westga.edu/~jhasbun/osp/Fourier.htm

[15] http://www.dspdimension.com/fftlab/

[16] http://www.academicearth.org/courses/the-fourier-transform-and-its-applications

[17] http:/

/

www.

encyclopediaofmath.

org/

index.

php?title=p/

f041150[18] http://mathworld.wolfram.com/FourierTransform.html

[19] http://www.dspdimension.com/admin/dft-a-pied/
http://www.dspdimension.com/admin/dft-a-pied/http://mathworld.wolfram.com/FourierTransform.htmlhttp://www.encyclopediaofmath.org/index.php?title=p/f041150http://www.academicearth.org/courses/the-fourier-transform-and-its-applicationshttp://www.dspdimension.com/fftlab/http://www.westga.edu/~jhasbun/osp/Fourier.htmhttp://www.thefouriertransform.com/http://www.nbtwiki.net/doku.php?id=tutorial:the_discrete_fourier_transformation_dfthttp://books.google.com/books?id=X-RFRHxMzvYC&pg=PA192&dq=%22The+Fourier+integral+can+be+regarded+as+an+extension+of+the+concept+of+Fourier+series%22&hl=en&sa=X&ei=D4rDT_vdCueQiAKF6PWeCA&ved=0CDQQ6AEwAA#v=onepage&q=%22The%20Fourier%20integral%20can%20be%20regarded%20as%20an%20extension%20of%20the%20concept%20of%20Fourier%20series%22&f=falsehttp://books.google.com/books?id=X-RFRHxMzvYC&pg=PA192&dq=%22The+Fourier+integral+can+be+regarded+as+an+extension+of+the+concept+of+Fourier+series%22&hl=en&sa=X&ei=D4rDT_vdCueQiAKF6PWeCA&ved=0CDQQ6AEwAA#v=onepage&q=%22The%20Fourier%20integral%20can%20be%20regarded%20as%20an%20extension%20of%20the%20concept%20of%20Fourier%20series%22&f=falsehttp://books.google.com/books?id=X-RFRHxMzvYC&pg=PA192&dq=%22The+Fourier+integral+can+be+regarded+as+an+extension+of+the+concept+of+Fourier+series%22&hl=en&sa=X&ei=D4rDT_vdCueQiAKF6PWeCA&ved=0CDQQ6AEwAA#v=onepage&q=%22The%20Fourier%20integral%20can%20be%20regarded%20as%20an%20extension%20of%20the%20concept%20of%20Fourier%20series%22&f=falsehttp://books.google.com/books?id=X-RFRHxMzvYC&pg=PA192&dq=%22The+Fourier+integral+can+be+regarded+as+an+extension+of+the+concept+of+Fourier+series%22&hl=en&sa=X&ei=D4rDT_vdCueQiAKF6PWeCA&ved=0CDQQ6AEwAA#v=onepage&q=%22The%20Fourier%20integral%20can%20be%20regarded%20as%20an%20extension%20of%20the%20concept%20of%20Fourier%20series%22&f=falsehttp://books.google.com/books?id=X-RFRHxMzvYC&pg=PA192&dq=%22The+Fourier+integral+can+be+regarded+as+an+extension+of+the+concept+of+Fourier+series%22&hl=en&sa=X&ei=D4rDT_vdCueQiAKF6PWeCA&ved=0CDQQ6AEwAA#v=onepage&q=%22The%20Fourier%20integral%20can%20be%20regarded%20as%20an%20extension%20of%20the%20concept%20of%20Fourier%20series%22&f=falsehttp://books.google.com/books?id=YUCV678MNAIC&dq=editions:xbArf-TFDSEC&source=gbs_navlinks_shttp://books.google.com/books?id=FAOc24bTfGkC&pg=PA158&dq=%22The+mathematical+thrust+of+the+principle%22&hl=en&sa=X&ei=Esa7T5PZIsqriQKluNjPDQ&ved=0CDQQ6AEwAA#v=onepage&q=%22The%20mathematical%20thrust%20of%20the%20principle%22&f=falsehttp://books.google.com/books?id=FAOc24bTfGkC&pg=PA158&dq=%22The+mathematical+thrust+of+the+principle%22&hl=en&sa=X&ei=Esa7T5PZIsqriQKluNjPDQ&ved=0CDQQ6AEwAA#v=onepage&q=%22The%20mathematical%20thrust%20of%20the%20principle%22&f=falsehttp://books.google.com/books?id=FAOc24bTfGkC&pg=PA158&dq=%22The+mathematical+thrust+of+the+principle%22&hl=en&sa=X&ei=Esa7T5PZIsqriQKluNjPDQ&ved=0CDQQ6AEwAA#v=onepage&q=%22The%20mathematical%20thrust%20of%20the%20principle%22&f=falsehttp://books.google.com/books?id=k_rdcKaUdr4C&pg=PA10http://books.google.com/books?id=tlLE4KUkk1gC&pg=PA256&dq=%22The+Fourier+transform+of+the+measure%22&hl=en&sa=X&ei=w8e7T43XJsiPiAKZztnRDQ&ved=0CEUQ6AEwAg#v=onepage&q=%22The%20Fourier%20transform%20of%20the%20measure%22&f=falsehttp://books.google.com/books?id=tlLE4KUkk1gC&pg=PA256&dq=%22The+Fourier+transform+of+the+measure%22&hl=en&sa=X&ei=w8e7T43XJsiPiAKZztnRDQ&ved=0CEUQ6AEwAg#v=onepage&q=%22The%20Fourier%20transform%20of%20the%20measure%22&f=falsehttp://books.google.com/books?id=tlLE4KUkk1gC&pg=PA256&dq=%22The+Fourier+transform+of+the+measure%22&hl=en&sa=X&ei=w8e7T43XJsiPiAKZztnRDQ&ved=0CEUQ6AEwAg#v=onepage&q=%22The%20Fourier%20transform%20of%20the%20measure%22&f=falsehttp://books.google.com/?id=QCcW1h835pwChttp://books.google.com/books?id=rfRnrhJwoloC&pg=PA29&dq=%22becomes+the+Fourier+%28integral%29+transform%22&hl=en&sa=X&ei=osO7T7eFOqqliQK3goXoDQ&ved=0CDQQ6AEwAA#v=onepage&q=%22becomes%20the%20Fourier%20%28integral%29%20transform%22&f=falsehttp://books.google.com/books?id=rfRnrhJwoloC&pg=PA29&dq=%22becomes+the+Fourier+%28integral%29+transform%22&hl=en&sa=X&ei=osO7T7eFOqqliQK3goXoDQ&ved=0CDQQ6AEwAA#v=onepage&q=%22becomes%20the%20Fourier%20%28integral%29%20transform%22&f=falsehttp://books.google.com/books?id=rfRnrhJwoloC&pg=PA29&dq=%22becomes+the+Fourier+%28integral%29+transform%22&hl=en&sa=X&ei=osO7T7eFOqqliQK3goXoDQ&ved=0CDQQ6AEwAA#v=onepage&q=%22becomes%20the%20Fourier%20%28integral%29%20transform%22&f=falsehttp://www.ams.org/mathscinet-getitem?mr=0262773http://dx.doi.org/10.1007%2Fs00041-012-9242-5http://books.google.com/books?id=-N8EAAAAYAAJ&pg=PA408&dq=%22that+is+to+say,+that+we+have+the+equation%22&hl=en&sa=X&ei=F667T-u5I4WeiALEwpHXDQ&ved=0CDgQ6AEwAA#v=onepage&q=%22that%20is%20to%20say%2C%20that%20we%20have%20the%20equation%22&f=falsehttp://books.google.com/books?id=-N8EAAAAYAAJ&pg=PA408&dq=%22that+is+to+say,+that+we+have+the+equation%22&hl=en&sa=X&ei=F667T-u5I4WeiALEwpHXDQ&ved=0CDgQ6AEwAA#v=onepage&q=%22that%20is%20to%20say%2C%20that%20we%20have%20the%20equation%22&f=falsehttp://books.google.com/books?id=-N8EAAAAYAAJ&pg=PA408&dq=%22that+is+to+say,+that+we+have+the+equation%22&hl=en&sa=X&ei=F667T-u5I4WeiALEwpHXDQ&ved=0CDgQ6AEwAA#v=onepage&q=%22that%20is%20to%20say%2C%20that%20we%20have%20the%20equation%22&f=falsehttp://books.google.com/books?id=TDQJAAAAIAAJ&pg=PA525&dq=%22c%27est-%C3%A0-dire+qu%27on+a+l%27%C3%A9quation%22&hl=en&sa=X&ei=SrC7T9yKBorYiALVnc2oDg&sqi=2&ved=0CEAQ6AEwAg#v=onepage&q=%22c%27est-%C3%A0-dire%20qu%27on%20a%20l%27%C3%A9quation%22&f=falsehttp://books.google.com/books?id=TDQJAAAAIAAJ&pg=PA525&dq=%22c%27est-%C3%A0-dire+qu%27on+a+l%27%C3%A9quation%22&hl=en&sa=X&ei=SrC7T9yKBorYiALVnc2oDg&sqi=2&ved=0CEAQ6AEwAg#v=onepage&q=%22c%27est-%C3%A0-dire%20qu%27on%20a%20l%27%C3%A9quation%22&f=falsehttp://books.google.com/books?id=TDQJAAAAIAAJ&pg=PA525&dq=%22c%27est-%C3%A0-dire+qu%27on+a+l%27%C3%A9quation%22&hl=en&sa=X&ei=SrC7T9yKBorYiALVnc2oDg&sqi=2&ved=0CEAQ6AEwAg#v=onepage&q=%22c%27est-%C3%A0-dire%20qu%27on%20a%20l%27%C3%A9quation%22&f=falsehttp://www.patternizando.com.br/2013/05/transformadas-discretas-wavelet-e-fourier-em-java/http://www.fourier-series.com/f-transform/index.htmlhttp://www.dspdimension.com/admin/dft-a-pied/http://en.wikipedia.org/w/index.php?title=MathWorldhttp://mathworld.wolfram.com/FourierTransform.htmlhttp://en.wikipedia.org/w/index.php?title=Eric_W._Weissteinhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/978-1-55608-010-4http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Springer_Science%2BBusiness_Mediahttp://en.wikipedia.org/w/index.php?title=Encyclopedia_of_Mathematicshttp://www.encyclopediaofmath.org/index.php?title=p/f041150http://www.academicearth.org/courses/the-fourier-transform-and-its-applicationshttp://www.dspdimension.com/fftlab/http://www.westga.edu/~jhasbun/osp/Fourier.htmhttp://www.thefouriertransform.com/http://www.nbtwiki.net/doku.php?id=tutorial:the_discrete_fourier_transformation_dfthttp://en.wikipedia.org/w/index.php?title=Special:BookSources/3-540-58654-7http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=K%C5%8Dsaku_Yosidahttp://en.wikipedia.org/w/index.php?title=Special:BookSources/0-471-30357-7http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Special:BookSources/978-0-8284-0324-5http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Edward_Charles_Titchmarsh


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[20] http://www.fourier-series.com/f-transform/index.html

[21] http://www.patternizando.com. br/2013/05/transformadas-discretas-wavelet-e-fourier-em-java/

Convolution

Visual comparison of convolution, cross-correlation and autocorrelation.

In mathematics and, in particular, functional

analysis, convolution is a mathematical

operation on two functions f and g,

producing a third function that is typically

viewed as a modified version of one of the

original functions, giving the area overlap

between the two functions as a function of

the amount that one of the original functions

is translated. Convolution is similar to

cross-correlation. It has applications that

include probability, statistics, computer

vision, image and signal processing,

electrical engineering, and differential equations.

The convolution can be defined for functions on groups other than Euclidean space. For example, periodic functions,

such as the discrete-time Fourier transform, can be defined on a circle and convolved byperiodic convolution. (See

row 10 at DTFT#Properties.) And discrete convolution can be defined for functions on the set of integers.

Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and

in the design and implementation of finite impulse response filters in signal processing.

Computing the inverse of the convolution operation is known as deconvolution.

Definition

The convolution off and g is writtenfg, using an asterisk or star. It is defined as the integral of the product of the

two functions after one is reversed and shifted. As such, it is a particular kind of integral transform:

(commutativity)

While the symbol t is used above, it need not represent the time domain. But in that context, the convolution formulacan be described as a weighted average of the function f() at the moment t where the weighting is given by g()

simply shifted by amount t. As t changes, the weighting function emphasizes different parts of the input function.

For functionsf, g defined on only, the integration domain is finite and the convolution is given by

In this case, the Laplace transform is more appropriate than the Fourier transform below and boundary terms become

relevant.

For the multi-dimensional formulation of convolution, see Domain of definition (below).
http://en.wikipedia.org/w/index.php?title=Integral_transformhttp://en.wikipedia.org/w/index.php?title=Asteriskhttp://en.wikipedia.org/w/index.php?title=Deconvolutionhttp://en.wikipedia.org/w/index.php?title=Finite_impulse_responsehttp://en.wikipedia.org/w/index.php?title=Numerical_linear_algebrahttp://en.wikipedia.org/w/index.php?title=Numerical_analysishttp://en.wikipedia.org/w/index.php?title=Integershttp://en.wikipedia.org/w/index.php?title=DTFT%23Propertieshttp://en.wikipedia.org/w/index.php?title=Circlehttp://en.wikipedia.org/w/index.php?title=Discrete-time_Fourier_transformhttp://en.wikipedia.org/w/index.php?title=Periodic_functionhttp://en.wikipedia.org/w/index.php?title=Euclidean_spacehttp://en.wikipedia.org/w/index.php?title=Group_%28mathematics%29http://en.wikipedia.org/w/index.php?title=Differential_equationshttp://en.wikipedia.org/w/index.php?title=Electrical_engineeringhttp://en.wikipedia.org/w/index.php?title=Signal_processinghttp://en.wikipedia.org/w/index.php?title=Image_processinghttp://en.wikipedia.org/w/index.php?title=Computer_visionhttp://en.wikipedia.org/w/index.php?title=Computer_visionhttp://en.wikipedia.org/w/index.php?title=Statisticshttp://en.wikipedia.org/w/index.php?title=Probabilityhttp://en.wikipedia.org/w/index.php?title=Cross-correlationhttp://en.wikipedia.org/w/index.php?title=Translation_%28geometry%29http://en.wikipedia.org/w/ind

integral transforms and delta function

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