Continuous Fourier transform

In mathematics, the continuous Fourier transform is a certain linear operator that maps function (mathematics)s to other functions. Loosely, the Fourier transform decomposes a function into a continuous spectrum of the frequencies that comprise that function. In mathematical physics, the Fourier transform of a signal

f(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 for a generalization.'') Suppose

f

is a complex-valued Lebesgue integration function. We then define its continuous Fourier transform

F

to be also a complex-valued function: :

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

for every real number

\omega

. (Here,

i

is the imaginary unit). We think of

\omega

as an angular frequency and

F(\omega)

as the complex number which is the amplitude and phase of the component of the signal

f(t)

at that frequency. The Fourier transform is close to a self-inverse mapping: if

F(\omega)

is defined as above, and

f

is sufficiently smooth, then :

f(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} F(\omega) e^{ i\omega t}\,d\omega

for every real number

t

. The factors

1\over\sqrt{2\pi}

before each integral are normalization factors. These factors are arbitrary so long as their product is equal to

1 \over 2 \pi

. The particular values chosen above are referred to as unitary normalization constants; another common choice is 1 and

1/2\pi

for the forward and inverse transforms, respectively. As a rule of thumb, mathematicians generally use the former (for symmetry reasons), while physicists and engineers use the latter. In addition, the Fourier coordinate

\omega

is sometimes replaced by

2 \pi \nu

, integrating over frequency

\nu

, in which case the unitary normalization constants are both equal to unity. Another arbitrary convention choice is whether the exponent is

+i\omega t

-i\omega t

in the forward transform—the only real requirement is that the forward- and inverse-transform exponents have opposite signs. ==Properties== === Completeness === If we define the Fourier transform

\mathcal{F}

in this way on the set of complex-valued functions on the line with compact support and Continuous linear extension to the Hilbert space of square-integrable functions, then it is a unitary operator :

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

Moreover, :

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

Note that in this relation, conjugation refers to the operator only, not to the entire Fourier transform of the function. ===Orthogonality=== The Fourier transform can also be defined for functions (and distributions) :

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

In the definition, the product

\omega t

is then to be interpreted as the inner product of the two vectors

\omega

and

t

. All the above properties and formulas remain valid. In this context, the functions

\frac{\exp \left(i \omega t \right)}{\sqrt{2 \pi}}

form an orthonormal basis in the space of tempered distributions :

\int_{-\infty}^\infty
 \left(\frac{e^{i\alpha t}}{\sqrt{2\pi}}\right)
 \,
 \left(\frac{e^{-i\beta t}}{\sqrt{2\pi}}\right)
 \,
 dt
=
\delta(\alpha - \beta).

The Fourier transform can be thought of as a transformation of coordinate basis in this space. ===The Plancherel theorem and 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 the Plancherel theorem: :

\int_{-\infty}^\infty f(t) g(t)^* \, dt =  \int_{-\infty}^\infty F(\omega) G(\omega)^* \, d\omega

(where the star denotes complex conjugation). Therefore, the Fourier transformation yields an Inner product space automorphism of the Hilbert space Lp space. Parseval's theorem, a special case of the Plancherel theorem, states that :

\int_{-\infty}^\infty \left| f(t) \right|^2 dt = \int_{-\infty}^\infty \left| F(\omega) \right|^2 d\omega.

This theorem is usually interpreted as asserting the unitary property of the Fourier transform. See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups. ===Localization property=== As a rule of thumb: the more concentrated

f(t)

is, the more spread out is

F(\omega)

. The only functions which coincide with their own Fourier transforms are the constant multiples of the function

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

. In a certain sense, this function therefore strikes the precise balance between being concentrated and being spread out. The Fourier transform also translates between smoothness and decay: if

f(t)

is several times differentiable, then

F(\omega)

decays rapidly towards zero for

s \to \plusmn \infin

. This can be more quantitatively expressed as follows. Suppose

f(t)

and

F(\omega)

are a Fourier transform pair. Without loss of generality, we can assume that

f(t)

is normalized: :

\int_{-\infty}^\infty f(t)f^*(t)\,dt=1.

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

\langle A\rangle \equiv \int_{-\infty}^\infty A(t)f(t)f^*(t)\,dt

and the expectation value of a function

B(\omega)

as: :

\langle B\rangle \equiv \int_{-\infty}^\infty B(\omega)F(t)F^*(\omega)\,d\omega

and then define the variance of

A(t)

as: :

\Delta^2 A\equiv\langle A^2-\langle A\rangle ^2\rangle

and similarly for the variance of

B(\omega)

, then it can be shown that :

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

The most famous practical example of this property is found in quantum mechanics. The momentum and position wave functions are Fourier transform pairs to within a factor of

h \over 2 \pi

and are normalized to unity. The above expression then becomes a statement of the Heisenberg uncertainty principle. ===Analysis of differential equations=== Fourier transforms, and the closely related Laplace transforms are widely used in solving differential equations. The Fourier transform is compatible with derivative 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 ''i''ω ''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 R^''n'' can also be translated into algebraic equations. ===Convolution theorem and cross-correlation theorem=== :''Main article:'' Convolution theorem The Fourier transform translates between convolution and multiplication of functions. If

f(t)

and

g(t)

are integrable functions with Fourier transforms

F(\omega)

and

G(\omega)

, respectively, and if the convolution of

f

and

g

exists and is integrable, then the Fourier transform of the convolution is given by the product of the Fourier transforms

F(\omega) G(\omega)

(possibly multiplied by a constant factor depending on the Fourier normalization convention). In the current normalization convention, this means that if :

h(t) = (f*g)(t) = \int_{-\infty}^\infty f(\tau)g(t - \tau)\,d\tau

then :

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

Also, if the product

f(t) g(t)

is integrable, then the Fourier transform of this product is given by the convolution of

F(\omega)

and

G(\omega)

, again with a constant factor. In the current normalization convention, this means that if :

h(t) = f(t) g(t)\,

then :

H(\omega) = \sqrt{2\pi} \int_{-\infty}^\infty F(\tau)G(\omega - \tau)\,d\tau.

In an analogous manner, it can be shown that if

h(t)

is the cross-correlation of

f(t)

and

g(t)

: :

h(t)=(f\star g)(t) = \int_{-\infty}^\infty f^*(\tau)\,g(t+\tau)\,d\tau

then the Fourier transform of

h(t)

is: :

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

where capital letters are again used to signify the Fourier transform. ===Tempered distributions=== The most general and natural context for studying the continuous Fourier transform is given by the distribution; 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 function is a tempered distribution but not a function; its Fourier transform is the constant function

1/\sqrt{2\pi}

. Distributions can be differentiated and the above mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions. If a function

f: \, \mathbb{R} \to \mathbb{C}

is square-integrable, that is :

\int_{-\infty}^\infty |f(t)|^2 \, dt < \infty,

then it can be viewed as a tempered distribution and hence has a Fourier transform. This transform is again square integrable. == Extension to higher dimensions == The Fourier transform can be extended to an N-dimensional space in a straightforward manner. If f(x) is a function of an N-dimensional vector x in the space, and k is the corresponding vector in the transform space (sometimes called the ''wavevector''), then :

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

where dx is an N-dimensional infinitesimal volume element in the space and the product in the exponential is the dot product. Using the N-dimensional orthogonality relationship: :

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

yields the inverse transform: :

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

==Table of important Fourier transforms== The following table records some important Fourier transforms.

F(\omega)

and

G(\omega)

denote the Fourier transforms of

f(t)

and

g(t)

, respectively.

f

and

g

may be integrable functions or tempered distributions. Note that this table's relations, and in particular constant factors such as

\sqrt{2\pi}

, depend upon the convention used for the Fourier transform definition above (although the general form of the relations is always the same). {| border="1" cellspacing=0 cellpadding="5" |- ! || Signal || Fourier transform || Remarks |- | 1 ||

a f(t) + b g(t)\,

a F(\omega) + b G(\omega)\,

|| Linearity |- | 2 ||

f(t - a)\,

e^{- i\omega a} F(\omega)\,

|| Shift in time domain |- | 3 ||

e^{ iat} f(t)\,

F(\omega - a)\,

|| Shift in frequency domain |- | 4 ||

f(a t)\,

|a|^{-1} F \left( \frac{\omega}{a} \right)\,

|| If

a\,

is large, then

f(a t)\,

is concentrated around 0 and

|a|^{-1}F(\frac{\omega}{a})\,

spreads out and flattens |- | 5 ||

\frac{d^n f(t)}{dt^n}\,

(i\omega)^n  F(\omega)\,

|| Generalized derivative property of the Fourier transform |- | 6 ||

t^n f(t)\,

i^n \frac{d^n F(\omega)}{d\omega^n}\,

|| This is the inverse rule to 5 |- | 7 ||

(f * g)(t)\,

\sqrt{2\pi} F(\omega) G(\omega)\,

f * g\,

denotes the convolution of

f\,

and

g\,

— this rule is the convolution theorem |- | 8 ||

f(t) g(t)\,

(F * G)(\omega) \over \sqrt{2\pi}\,

|| This is the inverse of 7 |- | 9 ||

\delta(t)\,

\frac{1}{\sqrt{2\pi}}\,

\delta(t)\,

denotes the Dirac delta distribution |- | 10 ||

1\,

\sqrt{2\pi}\delta(\omega)\,

|| Inverse of 9. This rule shows why the Dirac delta is important: it shows up as the Fourier transform of everyday functions |- | 11 ||

t^n\,

i^n \sqrt{2\pi} \delta^{(n)} (\omega)\,

|| Here,

n\,

is a natural number.

\delta^n(\omega)\,

is the ''n''-th distribution derivative of the Dirac delta. This rule follows from rules 6 and 10. Combining this rule with 1, we can transform all polynomials |- | 12 ||

e^{i a t}\,

\sqrt{2 \pi} \delta(\omega - a)\,

|| This follows from and 3 and 10 |- | 13 ||

\cos (a t)\,

\sqrt{2 \pi} \frac{\delta(\omega - a) + \delta(\omega + a)}{2}\,

|| Follows from rules 1 and 12 using

\cos(a t) = \frac{1}{2} \left( e^{i a t} + e^{-i a t}\right)\,

(Eulers formula in complex analysis) |- | 14 ||

\sin( at)\,

\sqrt{2 \pi}\frac{\delta(\omega - a) - \delta(\omega + a)}{2i}\,

|| Also from 1 and 12 |- | 15 ||

\exp(-a t^2)\,

\frac{1}{\sqrt{2a}} \exp\left(\frac{-\omega^2}{4a}\right)\,

|| Shows that the Gaussian function

\exp(-t^2/2)\,

is its own Fourier transform |- | 16 ||

W \sqrt{\frac{2}{\pi}} \mathrm{sinc}(W t)\,

\mathrm{rect}\left(\frac{\omega}{2 W}\right)\,

|| The rectangular function is a perfect low-pass filter and the sinc function is its time equivalent |- | 17 |

\frac{1}{t}\,

-i\sqrt{\frac{\pi}{2}}\sgn(\omega)\,

|Here

\sgn(\omega)\,

is the sign function; note that this is consistent with rules 6 and 10 |- | 18 |

\frac{1}{t^n}\,

-i\sqrt{\frac{\pi}{2}}\frac{(-i\omega)^{n-1}}{(n-1)!}\sgn(\omega)\,

|Generalization of rule 17 |- | 19 |

\sgn(t)\,

\sqrt{\frac{2}{\pi}}(i\omega)^{-1}\,

|The inverse of rule 17 |- | 20 |

\sqrt{2\pi}\mathrm{H}(t)\,

\frac{1}{i\omega} + \pi\delta(\omega)\,

|Here

\mathrm{H}(t)\,

is the Heaviside step function; this follows from rules 1 and 19 |} ==See also== *Fourier transform, *Fourier series, *Laplace transform, *Discrete Fourier transform. == External links== * [http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations. ==References== *[http://www.efunda.com/math/fourier_transform/ Fourier Transforms] from eFunda - includes tables * Dym & McKean, ''Fourier Series and Integrals''. (For readers with a background in mathematical analysis.) * K. Yosida, ''Functional Analysis'', Springer-Verlag, 1968. ISBN 3540586547 * L. H�rmander, ''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. Fourier analysis Integral transforms Unitary operators

Continuous Fourier transform

I think the table and the definition is not consistent -- there are some

2\pi

factors that don't tally. I won't fix it for fear of making it worse .... fiddly stuff. User:Lupin 15:16, 12 Jun 2004 (UTC) :It's a pain to keep things consistent through convention changes in the text, but nothing is jumping out at me right now. Which rules concern you? User:Stevenj 15:48, Jun 12, 2004 (UTC) ::There is at least an inconsistency with the Dirac

\delta

: ::"Furthermore, the useful Dirac delta is a tempered distribution but not a function; its Fourier transform is the constant function 1." ::Yet in the table the

\frac{1}{\sqrt{2\pi}}

convention is used. User:Eldacan 20:57, 6 Jul 2004 (UTC) :::The text is inconsistent with the FT definition used here; I'll correct it. User:Stevenj 04:04, Jul 7, 2004 (UTC) ==Convolution Theorem== ::Looks like I forgot about this page for a while :) Isn't the Fourier transform of a convolution the product of the fourier transforms, without any

2\pi

factors? Unless you want your convolutions to have constant factors too... this appears to need fixage on the convolution page as well. User:Lupin 10:21, 7 Jul 2004 (UTC) Whether you have constant factors in the convolution theorem depends upon your FT definition (and the convolution definition). With the FT and convolution definitions here, there are constant factors. Proof: Let: :

h(t) = \int_{-\infty}^{\infty} f(\tau) g(t - \tau) d\tau

f(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega

g(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} G(\omega) e^{i\omega t} d\omega

Then the Fourier transform of ''h'' is: :

H(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} h(t) e^{-i\omega t} dt

= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} dt \int_{-\infty}^{\infty} d\tau \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} d\omega'  F(\omega') e^{i\omega' \tau}  \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} d\omega'' G(\omega'') e^{i\omega'' (t-\tau)}  e^{-i\omega t}

= \frac{1}{\sqrt{2\pi}}  \int_{-\infty}^{\infty} d\omega'  F(\omega')  \int_{-\infty}^{\infty} d\omega'' G(\omega'')  \left( \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{i(\omega'' - \omega)t} dt \right)  \left( \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{i(\omega' - \omega'')t} d\tau \right)

= \frac{1}{\sqrt{2\pi}}  \int_{-\infty}^{\infty} d\omega'  F(\omega')  \int_{-\infty}^{\infty} d\omega'' G(\omega'')  \left( \sqrt{2\pi} \delta(\omega'' - \omega) \right)  \left( \sqrt{2\pi} \delta(\omega' - \omega'') \right)

= \sqrt{2\pi} F(\omega) G(\omega)

User:Stevenj 17:11, Jul 7, 2004 (UTC) ::Quite right, I dropped a constant in my rough calculation :-) By the way, there's a quicker proof where you don't need to use icky delta functions. The second line is obtained from the first with the change of variables

x'=x-t

H(y)=\frac{1}{\sqrt{2\pi}}\int\int f(t)g(x-t)\,dt\, e^{-ixy}\,dx

=\left(\frac{1}{\sqrt{2\pi}}\int f(t)e^{-ity}\,dt\right)\left(\frac{1}{\sqrt{2\pi}}\int g(x')\ e^{-ix'y}\,dx'\right)\cdot\sqrt{2\pi}

=F(y)G(y)\cdot\sqrt{2\pi}.

== Change normalization to simplify formulas? == Hello. I wonder if there's any support for changing the normalization convention to whatever makes the formulas come out simplest. For example, so that the convolution thm becomes F(f*g) = F(f) F(g). I believe that such a convention is commonly followed. Comments? User:Wile E. Heresiarch 02:02, 26 Oct 2004 (UTC) == main definition == I'm wondering why there is a

\frac{1}{\sqrt{2\pi}}

in front of the transform and inverse integrals. Most definitions that I've seen are:

X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-i\omega t}dt

and

x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(\omega) e^{i\omega t}d\omega

Ok, thanks! :As noted in the article, the normalization is somewhat arbitrary and varies between authors; the only important thing is that the product of the two constants is

1/2\pi

. (The one here is common among physicists and mathematicians; see e.g. ''Mathematical Methods for Physicists'' by Arfken and Weber. It has the nice property that the transform is unitary without scaling, so that the forward and backwards transforms are conjugates.) User:Stevenj 18:06, Nov 15, 2004 (UTC) --- I don't understand that: "The Fourier transform is close to a self-inverse mapping: if F(?) is defined as above, and f is sufficiently smooth, then..." if F(w) is defined as above - what does it refer to? ("as above") "f is sufficiently smooth" - what's the idea? f will be the result of the (inverse) transformation. How we can say before the transformation if f is sufficiently smooth? Jun 10, 2005 Typing four tildes (i.e, ~~~~) should do. User:Cburnett 05:17, Jun 11, 2005 (UTC)

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