Transformasi Fourier: Béda antarrépisi
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'''Transformasi Fourier''' nyéta hiji alat matematis anu ngawincik [[fungsi]] non-périodik kana fungsi-fungsi [[sinusoida]] anu nyusunna. Tranformasi Fourier ogé mangrupakeun alat pikeun ngarobah fungsi waktu kana wujud fungsi frékuénsi.
{{Transformasi Fourier}}▼
Dina [[matématika]],
==
Lamun x(t) mangrupakeun hiji sinyal non-
:<math>X(\omega) = \mathcal {F}\{x(t)\} = \int \limits _{-\infty}^{\infty} x(t)\ e^{-j \omega t}\,dt </math>
Kabalikan transformasi Fourier <math> X(\omega)</math> dilambangkeun ku <math> \mathcal {F^’}\{} </math> sarta didéfiniskieun kieu:
<math> x(t) = \mathcal {F^’}\{X(\omega)\} = \frac{1}{2\pi} \int _{-\infty}^{\infty} X(\omega)\ e^{ j\omega t}\,d\omega </math>
<math> x(t) jeung X(\omega)</math> disebut pasangan transformasi Fourier.
Urang ngagunakeun perlambang <math>x(t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad X(\omega)</math> pikeun ngalambangkeun yén ''x''(''t'') jeung ''X''(ω) mangrupakeun pasangan transformasi Fourier.
1. Liniéritas (superposisi):
::::<math>a\cdot x_1 (t) + b\cdot x_2 (t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad a\cdot X_1 (\omega) + b\cdot X_2 (\omega) </math>
2. Kakalian
::::{|
|<math>x_1 (t)\cdot x_2 (t) \,</math>
| <math>\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
\frac{1}{\sqrt{2\pi}}\cdot (X_1 * X_2)(\omega) \,</math>
| (konvensasi normalisasi uniter)
|-
|
| <math>\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
\frac{1}{2\pi}\cdot (X_1 * X_2 )(\omega) \,</math>
| (konvensi non-uniter)
|-
|
| <math>\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
(F*G)(\f) \,</math>
| (frekuensi biasa)
|}
3. Modulasi:
::::: <math>
\begin{align}
x(t)\cdot \cos \omega_{0}t
&\quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \frac{1}{2}[X(\omega+\omega_{0})+X(\omega-\omega_{0})],\qquad \omega_{0} \in \mathbb{R} \\
f(t)\cdot \sin \omega_{0}t
&\quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \frac{j}{2}[X(\omega+\omega_{0})-X(\omega-\omega_{0})] \\
x(t)\cdot e^{j\omega_{0}t}
&\quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad X(\omega-\omega_{0})
\end{align}
\,</math>
4. Géséran waktu
:<math>x(t – t_o) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad X(\omega)\ e^{-j \omega t_o} </math>
5. Géséran frékuénsi:
:<math>x(t) \ e^{j \omega_o t} \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad X(\omega – \omega_o) </math>
6. Skala:
::::<math> x(at) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \frac{1}{|a|}X\biggl(\frac{\omega}{a}\biggr), \qquad a \in \mathbb{R}, a \ne 0</math>
7. Lawan / kabalikan waktu:
::::<math>x(-t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad X(-\omega)</math>
8. Dualitas:
::::<math>X(t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad 2\pi x (-\omega)</math>
9. Diferensiasi waktu:
<math> x^’ (t) = \frac{d x(t)}{dt}\ x(t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad j\omega X(\omega) </math>
10. Diferensiasi frékuénsi:
<math> (-jt) x(t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad X^’ (\omega) = \frac{d X(\omega)}{dw} </math>
11. Integrasi:
::::<math>
\int_{-\infty}^{t} x(\tau)\, d\tau
\quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad
\frac{1}{j\omega}X(\omega)+\pi X(0)\cdot \delta(\omega),
\,</math>
==Transformasi Fourier tina sawatara sinyal nu mangfaat==
==Catetan==
{{reflist}}
==Tempo
*[[Dérét Fourier]]
*[[Transformasi Fourier gancang]] ''(Fast Fourier transform, FFT)''
Baris 24 ⟶ 107:
==Rujukan==
*[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]].)
Baris 33 ⟶ 115:
* R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed., Boston, McGraw Hill, 2000.
== Tumbu
* [http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations.
* {{MathWorld | urlname= FourierTransform | title= Fourier Transform}}
Baris 39 ⟶ 121:
* [http://www.ieee.li/pdf/viewgraphs_laplace.pdf Extending Laplace & Fourier Transforms by Dr. Shervin Erfani]
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[[ar:تحويل فوريي]]
Baris 74 ⟶ 155:
[[tr:Fourier dönüşümü]]
[[zh:傅里叶变换]]
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