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{{Transformasi Fourier}}
 
Dina [[matématika]], pikeun ngagéneralisasi réprésentasi [[Dérét Fourier|dérét Fourier]] <math>x(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_o t}.</math> sahingga bisa lumaku ogé pikeun sinyal non-périodik, maka digunakeun '''Transformasi Fourier'''.
Dina [[matématika]], '''transformasi Fourier''' atawa '''Fourier transform''' kontinyu ngarupakeun salasahiji wangunan husus tina analisa Fourier atawa [[analisis Fourier]]. Ku kituna, transformasi Fourier ngarobah hiji [[fungsi (matematika)|fungsi]] jadi fungsi séjénna, anu disebut simbul ''[[doméin frékuénsi]]'' ti fungsi asal (anu mindeng ngarupakeun fungsi dina [[doméin waktu]]). Dina hal husus ieu, kadua domein kontinyu sarta tanpa wates. Istilah [[transformasi Fourier]] bisa ngarujuk boh ka simbul domein frekuaensi tina sahiji fungsi atawa ka proses/rumus anu "ngarobah" hiji fungsi ka nu liana.
 
== Definisi ==
Lamun x(t) mangrupakeun hiji sinyal non-periodik. Mangka transformasi Fourier x(t), anu dilambangkeun ku <math>\mathcal{F}</math>, didefinisikeun ku
 
:<math>X(\omega) = <math>\mathcal{F}\{x(t)\}.\,</math> = \int \limits _{-\infty}^{\infty} x(t)\ e^{-j \omega t}\,dt, </math>
Aya sababaraha pajangjian umum pikeun ngadefinisikeun transformasi Fourier fungsi nu [[nomer komplék|boga harga kompleks]] [[integrasi Lebesgue|Lebesgue integrable]], <math>x.\,</math> &nbsp;Dina komunikasi katut [[pamrosesan sinyal]], sabage conto, inyana mindeng ngarupakeun fungsi''':'''
 
:<math>X(f) = \int \limits _{-\infty}^{\infty} x(t)\ e^{-i 2\pi f t}\,dt, </math> &nbsp; pikeun satiap [[angka nyata]] <math>f.\,</math>
 
Mangsa variable independen <math>t\,</math> ngalambangkeun ''waktu'' (kalayan [[SI]] unit [[waktu]]), variabel transformasi <math>f\,</math> ngalambangkeun [[frekuensi|frekuensi biasa]] (dina unit [[hertz]]). Lamun <math>x\,</math> ngarupakeun [[Hölder condition|Hölder kontinyu]], mangka inyana bisa direkonstruksi tina <math>X\,</math> ku cara kabalikan transformasi atawa '''inverse transform:'''
 
:<math>x(t) = \int \limits _{-\infty}^{\infty} X(f)\ e^{ i 2 \pi f t}\,df,</math> &nbsp; pikeun tiap angka ril <math>t.\,</math>
 
{{tarjamahkeun|Inggris}}
Other notations for <math>X(f)\,</math> are''':''' &nbsp;<math>\hat{x}(f)\,</math> &nbsp;and &nbsp;<math>\mathcal{F}\{x\}(f).\,</math>
 
The interpretation of <math>X\,</math> is aided by expressing it in [[polar coordinate]] form''':''' &nbsp;<math>X(f) = A(f)\ e^{i \phi (f)},\,</math> &nbsp;where''':'''
 
:<math>A(f) = |X(f)|, \, </math> &nbsp; the [[amplitudo]]
:<math>\phi (f) = \angle X(f), \, </math> &nbsp; the [[fase (gelombang)|fase]].
 
Then the inverse transform can be written''':'''
 
:<math>x(t) = \int \limits _{-\infty}^{\infty} A(f)\ e^{ i(2\pi f t +\phi (f))}\,df,</math>
 
which is a recombination of all the '''frequency components''' of <math>x(t).\,</math> &nbsp; Each component is a complex sinusoid of the form <math>e^{i 2\pi f t}</math> whose [[amplitudo]] is <math>A(f)</math> and whose initial [[phase angle]] (at <math>t=0</math>) is <math>\phi(f)</math>.
 
 
In [[mathematics]], the Fourier transform is commonly written in terms of [[angular frequency]]''':''' &nbsp;<math>\omega = 2\pi f,\,</math> &nbsp;whose units are [[radians]] per second.
 
The substitution <math>f = \frac{\omega}{2\pi}\,</math> into the formulas above produces this convention''':'''
 
:<math>X(\omega) = \int \limits _{-\infty}^\infty x(t)\ e^{- i\omega t}\,dt </math><ref>
''X''(''f'') and ''X''(&omega;) represent different, but related, functions, as shown in the table labeled ''Summary of popular forms of the Fourier transform''.
</ref>
 
:<math>x(t) = \frac{1}{2\pi} \int \limits _{-\infty}^{\infty} X(\omega)\ e^{ i\omega t}\,d\omega, </math>
 
which is also a [[Laplace_transform#Bilateral_Laplace_transform|bilateral Laplace transform]] evaluated at <math>s=i\omega</math>.
 
 
The <math>2\pi</math> factor can be split evenly between the Fourier transform and the inverse, which leads to another popular convention''':'''
 
:<math> X(\omega) = \frac{1}{\sqrt{2\pi}} \int \limits _{-\infty}^\infty x(t)\ e^{- i\omega t}\,dt </math>
 
:<math>x(t) = \frac{1}{\sqrt{2\pi}} \int \limits _{-\infty}^{\infty} X(\omega)\ e^{ i\omega t}\,d\omega. </math>
 
This convention and the <math>X(f)</math> convention are [[Unitary_operator|'''unitary''']] transforms.
 
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.
 
{| border="1" cellspacing="0" cellpadding="12"
|+ '''Summary of popular forms of the Fourier transform'''
|-
| rowspan="2" align="center" style="color: darkred"| '''angular <br /> frequency <br /> <math> \omega \, </math> <br />(rad/s)
| align="center" style="color: darkblue" | '''unitary'''
| <math> X_1(\omega) \ \stackrel{\mathrm{def}}{=}\ \frac{1}{\sqrt{2 \pi}} \int \limits _{-\infty}^{\infty} x(t) \ e^{-i \omega t}\, dt \ = \frac{1}{\sqrt{2 \pi}} X_2(\omega) = \frac{1}{\sqrt{2 \pi}} X_3 \left ( \frac{\omega}{2 \pi} \right )\,</math> <br />
<math> x(t) = \frac{1}{\sqrt{2 \pi}} \int \limits _{-\infty}^{\infty} X_1(\omega) \ e^{i \omega t}\, d \omega \ </math>
|-
| align="center" style="color: darkblue" | '''non-unitary'''
| <math> X_2(\omega) \ \stackrel{\mathrm{def}}{=}\ \int \limits _{-\infty}^{\infty} x(t) \ e^{-i \omega t} \ dt \ = \sqrt{2 \pi}\ X_1(\omega) = X_3 \left ( \frac{\omega}{2 \pi} \right ) \,</math> <br />
<math> x(t) = \frac{1}{2 \pi} \int \limits _{-\infty}^{\infty} X_2(\omega) \ e^{i \omega t} \ d \omega \ </math>
|-
| rowspan="2" align="center" style="color: darkred"| '''ordinary <br /> frequency <br /> <math> f \, </math> <br /> (hertz)
| align="center" style="color: darkblue" | '''unitary'''
| <math> X_3(f) \ \stackrel{\mathrm{def}}{=}\ \int \limits _{-\infty}^{\infty} x(t) \ e^{-i 2 \pi f t} \ dt \ = \sqrt{2 \pi}\ X_1(2 \pi f) = X_2(2 \pi f)\,</math><br />
<math> x(t) = \int \limits _{-\infty}^{\infty} X_3(f) \ e^{i 2 \pi f t}\, df \ </math>
|}
 
== Some Fourier transform properties ==
 
Notation''':''' <math>f(t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad F(\omega)</math> denotes that ''f''(''t'') and ''F''(&omega;) are a Fourier transform pair.
 
;Linearity
::::<math>a\cdot f(t) + b\cdot g(t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad a\cdot F(\omega) + b\cdot G(\omega) </math>
 
 
;Multiplication
::::{|
|<math>f(t)\cdot g(t) \,</math>
|&nbsp; &nbsp; <math>\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
\frac{1}{\sqrt{2\pi}}\cdot F(\omega) * G(\omega) \,</math>
| &nbsp; &nbsp; (unitary normalization convention)
|-
|
|&nbsp; &nbsp; <math>\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
\frac{1}{2\pi}\cdot F(\omega) * G(\omega) \,</math>
| &nbsp; &nbsp; (non-unitary convention)
|-
|
|&nbsp; &nbsp; <math>\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
F(f) * G(f) \,</math>
| &nbsp; &nbsp; (ordinary frequency)
|}
 
;e.g., Modulation
::::: <math>
\begin{align}
f(t)\cdot \cos \omega_{0}t
&\quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \frac{1}{2}[F(\omega+\omega_{0})+F(\omega-\omega_{0})],\qquad \omega_{0} \in \mathbb{R} \\
f(t)\cdot \sin \omega_{0}t
&\quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \frac{i}{2}[F(\omega+\omega_{0})-F(\omega-\omega_{0})] \\
f(t)\cdot e^{i\omega_{0}t}
&\quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad F(\omega-\omega_{0})
\end{align}
\,</math>
 
 
;Convolution
::::{|
|<math>f(t)* g(t) \,</math>
|&nbsp; &nbsp; <math>\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
\sqrt{2\pi}\cdot F(\omega)\cdot G(\omega) \,</math>
| &nbsp; &nbsp; (unitary convention)
|-
|
|&nbsp; &nbsp; <math>\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
F(\omega)\cdot G(\omega) \,</math>
| &nbsp; &nbsp; (non-unitary convention)
|-
|
|&nbsp; &nbsp; <math>\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
F(f)\cdot G(f) \,</math>
| &nbsp; &nbsp; (ordinary frequency)
|}
 
;e.g., Integration
::::<math>
f(t)*u(t)
= \int_{-\infty}^{t} f(\tau)\, d\tau
\quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
\frac{1}{i\omega}F(\omega)+\pi F(0)\cdot \delta(\omega)
\,</math>
 
 
;Conjugation
::::<math>\overline{f(t)} \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \overline{F(-\omega)}</math>
 
 
;Scaling
::::<math> f(at) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \frac{1}{|a|}F\biggl(\frac{\omega}{a}\biggr), \qquad a \in \mathbb{R}, a \ne 0</math>
 
 
;Time reversal
::::<math>f(-t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad F(-\omega)</math>
 
 
;Time shift
::::<math>f(t-t_0) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad e^{-i\omega t_0}\cdot F(\omega)</math>
 
 
;Parseval's theorem
::::{|
|<math>\int_{-\infty}^{\infty} f(t)\cdot \overline{g(t)}\, dt \,</math>
|<math>= \int_{-\infty}^{\infty} F(\omega)\cdot \overline{G(\omega)}\, d\omega \,</math>
| &nbsp; &nbsp; (unitary convention)
|-
|
|<math>= \frac{1}{2\pi}\cdot \int_{-\infty}^{\infty} F(\omega)\cdot \overline{G(\omega)}\, d\omega \,</math>
| &nbsp; &nbsp; (non-unitary convention)
|-
|
|<math>= \int_{-\infty}^{\infty} F(f)\cdot \overline{G(f)}\, df \,</math>
| &nbsp; &nbsp; (ordinary frequency)
|}
 
The section "Table of important Fourier transforms" (below) documents more properties of the continuous Fourier transform.
 
==Generalization==
 
Using two arbitrary real constants ''a'' and ''b'', the most general definition of the forward 1-dimensional Fourier transform is given by''':'''
 
:<math>X(\omega) = \sqrt{\frac{|b|}{(2 \pi)^{1-a}}} \int_{-\infty}^{+\infty} x(t) \cdot e^{-i b \omega t} \, dt, </math>
 
and the inverse is given by''':'''
 
:<math>x(t) = \sqrt{\frac{|b|}{(2 \pi)^{1+a}}} \int_{-\infty}^{+\infty} X(\omega) \cdot e^{i b \omega t} \, d\omega. </math>
 
Note that the transform definitions are symmetric; they can be reversed by simply changing the signs of ''a'' and ''b''.
 
The ordinary frequency convention corresponds to (''a'',''b'') = (0,2&pi;), and in that case the variable &omega; is changed to ''f''. &nbsp;If ''f'' and ''t'' carry units, their product must be dimensionless. For example, ''t'' may be in units of time, specifically [[second]]s, and ''f'' would be in [[hertz]].
 
The unitary, angular frequency convention is (''a'',''b'') = (0,1), and the non-unitary convention (above) is (''a'',''b'') = (1,1).
 
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 [[function composition|composition]] of the transform pair is defined to be the [[identity function|identity transformation]].
 
== More properties==
 
=== Completeness ===
 
We define the Fourier transform on the set of [[compact space|compactly]]-[[support (mathematics)|supported]] complex-valued functions of '''R''' and then [[Continuous linear extension|extend it by continuity]] to the [[Hilbert space]] of square-integrable functions with the usual inner-product. Then <math> \mathcal{F}</math>: ''L''<sup>2</sup>('''R''') &rarr; ''L''<sup>2</sup>('''R''') is a [[unitary operator]]. That is. <math>\mathcal{F}^*=\mathcal{F}^{-1}</math> and the transform preserves inner-products (see [[Parseval's theorem]], also described below). Note that, <math>\mathcal{F}^*</math> refers to [[Hermitian adjoint|adjoint]] of the Fourier Transform operator.
Moreover we can check that''':'''
 
:<math> \mathcal{F}^2 = \mathcal{J},\quad \mathcal{F}^3 = \mathcal{F}^* = \mathcal{F}^{-1}, \quad \mbox{and} \quad \mathcal{F}^4 = \mathcal{I}, </math>
 
where <math>\mathcal{J}</math> is the Time-Reversal operator defined as''':'''
 
:<math> \|\mathcal{J}\{f\}(t) - f(-t)\|_2 =0, </math>
 
and <math>\mathcal{I}</math> is the Identity operator defined as''':'''
 
:<math> \|\mathcal{I}\{f\}(t) - f(t)\|_2 =0.</math>
 
=== Multi-dimensional version ===
 
The Fourier transform, can be expanded to arbitrary dimension <math>n</math>. In the unitary, angular frequency convention, the definition is''':'''
 
:<math>F(\boldsymbol{\omega}) = \mathcal{F}_n\{f(\mathbf{x})\} \ \stackrel{\mathrm{def}}{=}\
\left(\frac{1}{\sqrt{2\pi}}\right)^{n}\int_{\R^n} f(\mathbf{x})\cdot e^{-i(\boldsymbol{\omega}\cdot \mathbf{x})}\,d\mathbf{x},</math>
 
where <math>\mathbf{x}</math> and <math>\boldsymbol{\omega}</math> are <math>n</math>-dimensional [[vector (mathematics)|vector]]s, and <math>\boldsymbol{\omega}\cdot \mathbf{x}</math> is the [[inner product]], also written <math>\left\langle \boldsymbol{\omega},\mathbf{x} \right\rangle,</math> of the 2 vectors. The integration is performed over all <math>n</math> dimensions.
 
<blockquote>
<math>f(\mathbf{x})</math> is assumed to belong to the "space" of integrable functions defined on '''R'''<sup>''n''</sup>''':'''
 
:<math> \mathcal{F}:L^1(\mathbb{R}^n)\rightarrow C(\mathbb{R}^n),</math>
 
where''':'''
 
:<math> L^1(\mathbb{R}^n) = \{f: \, \mathbb{R}^n \to \mathbb{C} \;\big|\; \int_{\mathbb{R}^n} |f(\mathbf{x})|\, d\mathbf{x} < \infty\},</math>
 
and ''C''('''R'''<sup>''n''</sup>) is the space of [[continuous function]]s on '''R'''<sup>''n''</sup>.
</blockquote>
 
One may now use this to define the continuous Fourier transform for compactly supported smooth functions, which are dense in ''L''<sup>2</sup>('''R'''<sup>n</sup>). The [[Plancherel theorem]] then allows us to extend the definition of the Fourier transform to functions on ''L''<sup>2</sup>('''R'''<sup>''n''</sup>) (even those not compactly supported) by continuity arguments. All the properties and formulas listed on this page apply to the Fourier transform so defined.
 
Unfortunately, further extensions become more technical. One may use the [[Hausdorff-Young inequality]] to define the Fourier transform for ''f'' &isin; ''L''<sup>''p''</sup>('''R'''<sup>''n''</sup>) for 1 &le; ''p'' &le; 2. The Fourier transform of functions in ''L''<sup>''p''</sup> for the range 2 < ''p'' < &infin; requires the study of distributions, since the Fourier transform of some functions in these spaces is no longer a function, but rather a [[distribution (mathematics)|distribution]].
 
===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''(&omega;) and ''G''(&omega;) are their unitary Fourier transforms, then we have [[Parseval's theorem]]''':'''
 
: <math>\int_{\mathbb{R}^n} f(t) \bar{g}(t) \, dt = \int_{\mathbb{R}^n} F(\omega) \bar{G}(\omega) \, d\omega,</math>
 
where the bar denotes [[complex conjugation]]. Therefore, the Fourier transformation yields an [[Inner product space|isometric]] [[automorphism]] of the [[Hilbert space]] ''L''<sup>2</sup>('''R'''<sup>n</sup>).
 
The [[Plancherel theorem]], which is equivalent to [[Parseval's theorem]], states''':'''
 
:<math>\int_{\mathbb{R}^n} \left| f(t) \right|^2\, dt = \int_{\mathbb{R}^n} \left| F(\omega) \right|^2\, d\omega. </math>
 
This theorem is usually interpreted as asserting the [[unitary operator|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===<!-- This section is linked from [[Laser]] -->
 
As a rule of thumb: the more concentrated ''f''(''t'') is, the more spread out ''F''(&omega;) is. In particular, if we "squeeze" a function in ''t'', it spreads out in &omega; 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]]
 
:<math>f(t) = \exp \left( \frac{-t^2}{2} \right).</math>
 
This function is related to [[Gaussian distribution]]s, and in fact, is an [[eigenfunction]] of the Fourier transform operators. Again, it is worth
stressing that the mere fact that the Gaussian is self-dual does not make it in any way special: many self-dual functions exist.
 
The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of a '''Fourier Uncertainty Principle'''. Suppose ''f''(''t'') and ''F''(&omega;) are a Fourier transform pair for a finite-energy (i.e. square-integrable) function. Without loss of generality, we assume that ''f''(''t'') is normalized:
 
:<math>\int_{-\infty}^\infty |f(t)|^2 \,dt=1.</math>
 
It follows from Parseval's theorem that ''F''(&omega;) is also normalized.
 
Define the [[expected value|expected]] ''location''<ref name = HUP>Location, momentum and particle do not have any physical meaning here; they are simply convenient monikers chosen with analogy to the interpretation used in the Heisenberg Uncertainty Principle. </ref> of a particle (with probability density |''f''(''t)''|<sup>2</sup>) as
 
:<math>u_f \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^\infty t|f(t)|^2\,dt.</math>
 
and the expectation value of the ''momentum''<ref name=HUP/> of the particle (with probability density |''f''(&omega;)|<sup>2</sup>) as
 
:<math>\xi_F \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^\infty \omega |F(\omega)|^2\,d\omega.</math>
 
Also define the [[variance]]s around the above-defined average values as
 
:<math>\sigma^2_{f} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^\infty (t-u_f)^2|f(t)|^2\,dt </math>
 
and
 
:<math>\sigma^2_{F} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^\infty (\omega-\xi_F)^2 |F(\omega)|^2\,d\omega. </math>
 
Then it can be shown that
 
:<math>\sigma^2_{f}\, \sigma^2_{F} \ge \frac{1}{4}.</math>
 
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]]. Following from the axioms of quantum mechanics, the momentum and position wave functions are Fourier transform pairs to within a factor of ''h''/2&pi; and are normalized to unity. The above expression then becomes a statement of the [[Heisenberg uncertainty principle]].
 
The Fourier transform also translates between smoothness and decay. If ''f''(''t'') is several times differentiable, then ''F''(&omega;) decays rapidly towards zero for &omega; &rarr; &plusmn; &infin;.
 
===Analysis of differential equations===
 
Fourier transforms, and the closely related [[Laplace transform]]s are widely used in solving [[differential equations]]. The Fourier transform is compatible with [[derivative|differentiation]] in the following sense: if ''f''(''t'') is a differentiable function with Fourier transform ''F''(&omega;), then the Fourier transform of its derivative is given by ''i''&omega; ''F''(&omega;). 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 equation]]s with domain '''R'''<sup>n</sup> can also be translated into algebraic equations.
 
===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''(&omega;) and ''H''(&omega;) 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''(&omega;) ''H''(&omega;) (possibly multiplied by a constant factor depending on the Fourier normalization convention).
 
In the unitary normalization convention, this means that if''':'''
 
:<math>g(t) = \{f*h\}(t) = \int_{-\infty}^\infty f(\tau)h(t - \tau)\,d\tau,</math>
 
where * denotes the convolution operation, then''':'''
 
:<math>G(\omega) = \sqrt{2\pi}\cdot F(\omega)H(\omega).\,</math>
 
The above formulas hold true for functions defined on both one- and multi-dimension real space. In [[LTI system theory|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 [[Dirac delta function|unit impulse]] for ''f''(''t'') yields ''g''(''t'')=''h''(''t''). In this case, ''H''(&omega;) represents the [[frequency response]] of the system.
 
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''(&omega;) and ''Q''(&omega;), again with a constant scaling factor.
 
In the unitary normalization convention, this means that if ''f''(''t'') = ''p''(''t'') ''q''(''t'') then:
 
:<math>F(\omega) = \frac{1}{\sqrt{2\pi}} \bigg( P(\omega) * Q(\omega) \bigg) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty P(\alpha)Q(\omega - \alpha)\,d\alpha.</math>
 
=== Cross-correlation theorem ===
 
In an analogous manner, it can be shown that if <math>g(t)</math> is the [[cross-correlation]] of <math>f(t)</math> and <math>h(t)</math>:
 
:<math>g(t)=(f\star h)(t) = \int_{-\infty}^\infty \bar{f}(\tau)\,h(t+\tau)\,d\tau</math>
 
then the Fourier transform of <math>g(t)</math> is:
 
:<math>G(\omega) = \sqrt{2\pi}\,\overline{F}(\omega)\,H(\omega)</math>
 
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 [[distribution (mathematics)#Tempered distributions and Fourier transform|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 function|Dirac delta]] is a tempered distribution but not a function; its Fourier transform is a constant function (whose specific value depends upon the form of the Fourier transform used). Distributions can be differentiated and the above mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.
 
==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===
{| class="wikitable"
! !! Signal !! Fourier transform <br /> unitary, angular frequency !! Fourier transform <br /> unitary, ordinary frequency !! Remarks
|-
|
|align="center"|<math> g(t)\,</math>
|align="center"|<math> G(\omega)\!\ \stackrel{\mathrm{def}}{=}\ \!</math><br /><br /><math>\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\!\!g(t) e^{-i \omega t}\, dt </math>
|align="center"|<math> G(f)\!\ \stackrel{\mathrm{def}}{=}\ </math><br /><br /><math>\int_{-\infty}^{\infty}\!\!g(t) e^{-i 2\pi f t}\, dt </math>
|
|-
| 101
|<math>a\cdot g(t) + b\cdot h(t)\,</math>
|<math>a\cdot G(\omega) + b\cdot H(\omega)\,</math>
|<math>a\cdot G(f) + b\cdot H(f)\,</math>
|Linearity
|-
| 102
|<math>g(t - a)\,</math>
|<math>e^{- i a \omega} G(\omega)\,</math>
|<math>e^{- i 2\pi a f} G(f)\,</math>
|Shift in time domain
|-
| 103
|<math>e^{ iat} g(t)\,</math>
|<math>G(\omega - a)\,</math>
|<math>G \left(f - \frac{a}{2\pi}\right)\,</math>
|Shift in frequency domain, dual of 102
|-
| 104
|<math>g(a t)\,</math>
|<math>\frac{1}{|a|} G \left( \frac{\omega}{a} \right)\,</math>
|<math>\frac{1}{|a|} G \left( \frac{f}{a} \right)\,</math>
|If <math>|a|\,</math> is large, then <math>g(a t)\,</math> is concentrated around 0 and <math>\frac{1}{|a|}G \left( \frac{\omega}{a} \right)\,</math> spreads out and flattens. It is interesting to consider the limit of this as <math>|a|</math> tends to infinity - the delta function
|-
| 105
|<math>G(t)\,</math>
|<math> g(-\omega)\,</math>
|<math> g(-f)\,</math>
|Duality property of the Fourier transform. Results from swapping "dummy" variables of <math> t \,</math> and <math> \omega \,</math>.
|-
| 106
|<math>\frac{d^n g(t)}{dt^n}\,</math>
|<math> (i\omega)^n G(\omega)\,</math>
|<math> (i 2\pi f)^n G(f)\,</math>
|Generalized derivative property of the Fourier transform
|-
| 107
|<math>t^n g(t)\,</math>
|<math>i^n \frac{d^n G(\omega)}{d\omega^n}\,</math>
|<math>\left (\frac{i}{2\pi}\right)^n \frac{d^n G(f)}{df^n}\,</math>
|This is the dual of 106
|-
| 108
|<math>(g * h)(t)\,</math>
|<math>\sqrt{2\pi} G(\omega) H(\omega)\,</math>
|<math>G(f) H(f)\,</math>
|<math>g * h\,</math> denotes the [[convolution]] of <math>g\,</math> and <math>h\,</math> &mdash; this rule is the [[convolution theorem]]
|-
| 109
|<math>g(t) h(t)\,</math>
|<math>(G * H)(\omega) \over \sqrt{2\pi}\,</math>
|<math>(G * H)(f)\,</math>
|This is the dual of 108
|-
| 110
|<math>g(t)\,</math> is purely real, and an [[even function]]
|colspan="2" align="center"|<math>G(\omega)\,</math> and <math>G(f)\,</math> are purely real, and [[even function]]s
|
|-
| 111
|<math>g(t)\,</math> is purely real, and an [[odd function]]
|colspan="2" align="center"|<math>G(\omega)\,</math> and <math>G(f)\,</math> are purely [[imaginary number|imaginary]], and [[odd function]]s
|
|}
 
===Square-integrable functions===
{| class="wikitable"
! !! Signal !! Fourier transform <br /> unitary, angular frequency !! Fourier transform <br /> unitary, ordinary frequency !! Remarks
|-
|
|align="center"|<math> g(t) \,</math>
|align="center"|<math> G(\omega)\!\ \stackrel{\operatorname{def}}{=}\ \!</math><br /><br /><math>\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\!\!g(t) e^{-i \omega t} \operatorname{d}t \,</math>
|align="center"|<math> G(f)\!\ \stackrel{\operatorname{def}}{=}\ </math><br /><br /><math>\int_{-\infty}^{\infty}\!\!g(t) e^{-i 2\pi f t} \operatorname{d}t \,</math>
|
|-
| 201
|<math>\operatorname{rect}(a t) \,</math>
|<math>\frac{1}{\sqrt{2 \pi a^2}}\cdot \operatorname{sinc}\left(\frac{\omega}{2\pi a}\right)</math>
|<math>\frac{1}{|a|}\cdot \operatorname{sinc}\left(\frac{f}{a}\right)</math>
|The [[rectangular function|rectangular pulse]] and the ''normalized'' [[sinc function]], here defined as <math>\operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x}</math>
|-
| 202
|<math> \operatorname{sinc}(a t)\,</math>
|<math>\frac{1}{\sqrt{2\pi a^2}}\cdot \operatorname{rect}\left(\frac{\omega}{2 \pi a}\right)</math>
|<math>\frac{1}{|a|}\cdot \operatorname{rect}\left(\frac{f}{a} \right)\,</math>
|Dual of rule 201. The [[rectangular function]] is an idealized [[low-pass filter]], and the [[sinc function]] is the [[acausal|non-causal]] impulse response of such a filter.
|-
| 203
|<math> \operatorname{sinc}^2 (a t) \,</math>
|<math> \frac{1}{\sqrt{2\pi a^2}}\cdot \operatorname{tri} \left( \frac{\omega}{2\pi a} \right) </math>
|<math> \frac{1}{|a|}\cdot \operatorname{tri} \left( \frac{f}{a} \right) </math>
| ''tri'' is the [[triangular function]]
|-
| 204
|<math> \operatorname{tri} (a t) \,</math>
|<math>\frac{1}{\sqrt{2\pi a^2}} \cdot \operatorname{sinc}^2 \left( \frac{\omega}{2\pi a} \right) </math>
|<math>\frac{1}{|a|}\cdot \operatorname{sinc}^2 \left( \frac{f}{a} \right) \,</math>
| Dual of rule 203.
|-
| 205
|<math>e^{-\alpha t^2}\,</math>
|<math>\frac{1}{\sqrt{2 \alpha}}\cdot e^{-\frac{\omega^2}{4 \alpha}}</math>
|<math>\sqrt{\frac{\pi}{\alpha}}\cdot e^{-\frac{(\pi f)^2}{\alpha}}</math>
|Shows that the [[Gaussian function]] <math>\exp(-\alpha t^2)</math> is its own Fourier transform. For this to be integrable we must have <math>\operatorname{Re}(\alpha)>0</math>.
|-
| 206
|<math> e^{iat^2} = \left. e^{-\alpha t^2}\right|_{\alpha = -i a} \,</math>
|<math> \frac{1}{\sqrt{2 a}} \cdot e^{-i \left(\frac{\omega^2}{4 a} -\frac{\pi}{4}\right)}</math>
|<math> \sqrt{\frac{\pi}{a}} \cdot e^{-i \left(\frac{\pi^2 f^2}{a} -\frac{\pi}{4}\right)} </math>
| common in [[optics]]
|-
| 207
|<math>\cos ( a t^2 ) \,</math>
|<math> \frac{1}{\sqrt{2 a}} \cos \left( \frac{\omega^2}{4 a} - \frac{\pi}{4} \right) </math>
|<math> \sqrt{\frac{\pi}{a}} \cos \left( \frac{\pi^2 f^2}{a} - \frac{\pi}{4} \right) </math>
|
|-
| 208
|<math>\sin ( a t^2 ) \,</math>
|<math> \frac{-1}{\sqrt{2 a}} \sin \left( \frac{\omega^2}{4 a} - \frac{\pi}{4} \right) </math>
|<math> - \sqrt{\frac{\pi}{a}} \sin \left( \frac{\pi^2 f^2}{a} - \frac{\pi}{4} \right) </math>
|
|-
| 209
|<math>\operatorname{e}^{-a|t|} \,</math>
|<math> \sqrt{\frac{2}{\pi}} \cdot \frac{a}{a^2 + \omega^2} </math>
|<math> \frac{2 a}{a^2 + 4 \pi^2 f^2} </math>
| ''a>0''
|-
| 210
|<math> \frac{1}{\sqrt{|t|}} \,</math>
|<math> \frac{1}{\sqrt{|\omega|}}</math>
|<math> \frac{1}{\sqrt{|f|}} </math>
| the transform is the function itself
|-
| 211
|<math> J_0 (t)\,</math>
|<math> \sqrt{\frac{2}{\pi}} \cdot \frac{\operatorname{rect} \left( \frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}} </math>
|<math> \frac{2\cdot \operatorname{rect} (\pi f)}{\sqrt{1 - 4 \pi^2 f^2}} </math>
| ''J<sub>0</sub>(t)'' is the [[Bessel function]] of first kind of order 0
|-
| 212
|<math> J_n (t) \,</math>
|<math> \sqrt{\frac{2}{\pi}} \frac{ (-i)^n T_n (\omega) \operatorname{rect} \left( \frac{\omega}{2} \right)}{\sqrt{1 - \omega^2}} </math>
|<math> \frac{2 (-i)^n T_n (2 \pi f) \operatorname{rect} (\pi f)}{\sqrt{1 - 4 \pi^2 f^2}} </math>
| it's the generalization of the previous transform; ''T<sub>n</sub> (t)'' is the [[Chebyshev polynomials|Chebyshev polynomial of the first kind]].
|-
| 213
|<math> \frac{J_n (t)}{t} \,</math>
|<math> \sqrt{\frac{2}{\pi}} \frac{i}{n} (-i)^n \cdot U_{n-1} (\omega)\,</math><br>
&nbsp; <math>\cdot \ \sqrt{1 - \omega^2} \operatorname{rect} \left( \frac{\omega}{2} \right) </math>
|<math> \frac{2 i}{n} (-i)^n \cdot U_{n-1} (2 \pi f)\,</math><br>
&nbsp; <math>\cdot \ \sqrt{1 - 4 \pi^2 f^2} \operatorname{rect} ( \pi f ) </math>
| ''U<sub>n</sub> (t)'' is the [[Chebyshev polynomials|Chebyshev polynomial of the second kind]]
|-
| 214
|<math>\operatorname{sech}(a t) \,</math>
|<math>\frac{1}{a}\sqrt{\frac{\pi}{2}}\operatorname{sech} \left( \frac{\pi}{2 a} \omega \right)</math>
|<math>\frac{\pi}{a} \operatorname{sech} \left( \frac{\pi^2}{ a} f \right)</math>
|[[Hyperbolic function|Hyperbolic secant]] is its own Fourier transform
|}
 
===Distributions===
{| class="wikitable"
! !! Signal !! Fourier transform <br /> unitary, angular frequency !! Fourier transform <br /> unitary, ordinary frequency !! Remarks
|-
|
|align="center"|<math> g(t) \,</math>
|align="center"|<math> G(\omega)\!\ \stackrel{\mathrm{def}}{=}\ \!</math><br /><br /><math>\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\!\!g(t) e^{-i \omega t}\, dt </math>
|align="center"|<math> G(f)\!\ \stackrel{\mathrm{def}}{=}\ </math><br /><br /><math>\int_{-\infty}^{\infty}\!\!g(t) e^{-i 2\pi f t}\, dt </math>
|
|-
| 301
|<math>1\,</math>
|<math>\sqrt{2\pi}\cdot \delta(\omega)\,</math>
|<math>\delta(f)\,</math>
|<math>\displaystyle\delta(\omega)</math> denotes the [[Dirac delta]] distribution.
|-
| 302
|<math>\delta(t)\,</math>
|<math>\frac{1}{\sqrt{2\pi}}\,</math>
|<math>1\,</math>
|Dual of rule 301.
|-
| 303
|<math>e^{i a t}\,</math>
|<math>\sqrt{2 \pi}\cdot \delta(\omega - a)\,</math>
|<math>\delta\left(f - \frac{a}{2\pi}\right)\,</math>
|This follows from 103 and 301.
|-
| 304
|<math>\cos (a t)\,</math>
|<math>\sqrt{2 \pi} \frac{\delta(\omega\!-\!a)\!+\!\delta(\omega\!+\!a)}{2}\,</math>
|<math>\frac{\delta(f\!-\!\begin{matrix}\frac{a}{2\pi}\end{matrix})\!+\!\delta(f\!+\!\begin{matrix}\frac{a}{2\pi}\end{matrix})}{2}\,</math>
|Follows from rules 101 and 303 using [[Eulers formula in complex analysis|Euler's formula]]: <math>\displaystyle\cos(a t) = (e^{i a t} + e^{-i a t})/2.</math>
|-
| 305
|<math>\sin( at)\,</math>
|<math>i \sqrt{2 \pi}\frac{\delta(\omega\!+\!a)\!-\!\delta(\omega\!-\!a)}{2}\,</math>
|<math>i \frac{\delta(f\!+\!\begin{matrix}\frac{a}{2\pi}\end{matrix})\!-\!\delta(f\!-\!\begin{matrix}\frac{a}{2\pi}\end{matrix})}{2}\,</math>
|Also from 101 and 303 using <math>\displaystyle\sin(a t) = (e^{i a t} - e^{-i a t})/(2i).</math>
|-
| 306
|<math>t^n\,</math>
|<math>i^n \sqrt{2\pi} \delta^{(n)} (\omega)\,</math>
|<math>\left(\frac{i}{2\pi}\right)^n \delta^{(n)} (f)\,</math>
|Here, <math>\displaystyle n</math> is a [[natural number]]. <math>\displaystyle\delta^{(n)}(\omega)</math> is the <math>\displaystyle n</math>-th distribution derivative of the Dirac delta. This rule follows from rules 107 and 302. Combining this rule with 1, we can transform all [[polynomial]]s.
|-
| 307
|<math>\frac{1}{t}\,</math>
|<math>-i\sqrt{\frac{\pi}{2}}\sgn(\omega)\,</math>
|<math>-i\pi\cdot \sgn(f)\,</math>
|Here <math>\displaystyle\sgn(\omega)</math> is the [[sign function]]; note that this is consistent with rules 107 and 302.
|-
| 308
|<math>\frac{1}{t^n}\,</math>
|<math>-i \begin{matrix} \sqrt{\frac{\pi}{2}}\cdot \frac{(-i\omega)^{n-1}}{(n-1)!}\end{matrix} \sgn(\omega)\,</math>
|<math>-i\pi \begin{matrix} \frac{(-i 2\pi f)^{n-1}}{(n-1)!}\end{matrix} \sgn(f)\,</math>
|Generalization of rule 307.
|-
| 309
|<math>\sgn(t)\,</math>
|<math>\sqrt{\frac{2}{\pi}}\cdot \frac{1}{i\ \omega }\,</math>
|<math>\frac{1}{i\pi f}\,</math>
|The dual of rule 307.
|-
| 310
|<math> u(t) \,</math>
|<math>\sqrt{\frac{\pi}{2}} \left( \frac{1}{i \pi \omega} + \delta(\omega)\right)\,</math>
|<math>\frac{1}{2}\left(\frac{1}{i \pi f} + \delta(f)\right)\,</math>
|Here <math>u(t)</math> is the Heaviside [[Heaviside step function|unit step function]]; this follows from rules 101 and 309.
|-
| 311
|<math> e^{- a t} u(t) \,</math>
|<math>\frac{1}{\sqrt{2 \pi} (a + i \omega)}</math>
|<math>\frac{1}{a + i 2 \pi f}</math>
|<math>u(t)</math> is the Heaviside [[Heaviside step function|unit step function]] and <math>a > 0</math>.
|-
| 312
|<math>\sum_{n=-\infty}^{\infty} \delta (t - n T) \,</math>
|<math>\begin{matrix} \frac{\sqrt{2\pi }}{T}\end{matrix} \sum_{k=-\infty}^{\infty} \delta \left( \omega -k \begin{matrix} \frac{2\pi }{T}\end{matrix} \right)\,</math>
|<math>\frac{1}{T} \sum_{k=-\infty}^{\infty} \delta \left( f -\frac{k }{T}\right) \,</math>
|The [[Dirac comb]] &mdash; helpful for explaining or understanding the transition from continuous to discrete time.
|}
 
== About notation ==
The Fourier transform is a mapping on a function space. This mapping is here denoted <math>\mathcal{F}</math> and <math>\mathcal{F}\{s\}</math> is used to denote the Fourier transform of the function ''s''. This mapping is linear, which means that <math>\mathcal{F}</math> can also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra of applying a linear transformation to a vector (here the signal ''s'') can be used to write <math>\mathcal{F} s</math> instead of <math>\mathcal{F}\{s\}</math>. 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 <math>\omega</math> for its variable, and this is denoted either as <math>\mathcal{F}\{s\}(\omega)</math> or as <math>(\mathcal{F} s)(\omega)</math>. Notice that in the former case, it is implicitly understood that <math>\mathcal{F}</math> is applied first to ''s'' and then the resulting function is evaluated at <math>\omega</math>, not the other way around.
 
In mathematics and various applied sciences it is often necessary distinguish between a function ''s'' and the value of ''s'' when its variable equals ''t'', denoted ''s(t)''. This means that a notation like <math>\mathcal{F}\{s(t)\}</math> formally can be interpreted as the Fourier transform of the values of ''s'' at ''t'', which must be considered as an ill-formed expression since it describes the Fourier transform of a function value rather than of a function. 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, <math>\mathcal{F}\{ \mathrm{rect}(t) \} = \mathrm{sinc}(\omega)</math> is sometimes used to express that the Fourier transform of a rectangular function is a sinc function, or <math>\mathcal{F}\{s(t+t_{0})\} = \mathcal{F}\{s(t)\} e^{i \omega t_{0}}</math> 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 of ''t'', not of <math>t_{0}</math>. If possible, this informal usage of the <math>\mathcal{F}</math> operator should be avoided, in particular when it is not perfectly clear which variable the function to be transformed depends on.
 
==Catetan==
Dicomot ti "https://su.wikipedia.org/wiki/Transformasi_Fourier"