Transformasi Fourier

Transformasi Fourier nyéta hiji alat matematis anu ngawincik fungsi non-périodik kana fungsi-fungsi sinusoida anu nyusunna. Tranformasi Fourier ogé mangrupa alat pikeun ngarobah fungsi waktu kana wujud fungsi frékuénsi.

Dina matématika, lamun fungsi périodik bisa diwincik kana sajumlah dérét fungsi anu disebut deret Fourier ku rumus ${\displaystyle x(t)=\sum _{n=-\infty }^{\infty }c_{n}e^{jn\omega _{o}t}.}$ mangka géneralisasi pikeun fungsi non-périodik bisa dilakukeun maké rumus nu disebut transformasi Fourier. Jadi transformasi Fourier mangrupa generalisasi tina dérét Fourier

Définisi

Lamun x(t) mangrupa hiji sinyal non-périodik. Mangka transformasi Fourier x(t), anu dilambangkeun ku ${\displaystyle {\mathcal {F}}}$, didéfinisikeun ku

${\displaystyle X(\omega )={\mathcal {F}}\{x(t)\}=\int \limits _{-\infty }^{\infty }x(t)\ e^{-j\omega t}\,dt}$

Kabalikan transformasi Fourier ${\displaystyle X(\omega )}$ dilambangkeun ku ${\displaystyle {\mathcal {F^{'}}}}$ sarta didéfiniskieun kieu:

${\displaystyle x(t)={\mathcal {F}}^{'}\{X(\omega )\}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )\ e^{j\omega t}\,d\omega ,}$   pikeun tiap angka ril t.

di mana ${\displaystyle x(t)jeungX(\omega )}$ disebut pasangan transformasi Fourier.

Sifat Transformasi Fourier

Urang ngagunakeun perlambang ${\displaystyle x(t)\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad X(\omega )}$ pikeun ngalambangkeun yén x(t) jeung X(ω) mangrupa pasangan transformasi Fourier.

1. Liniéritas (superposisi):

${\displaystyle a\cdot x_{1}(t)+b\cdot x_{2}(t)\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad a\cdot X_{1}(\omega )+b\cdot X_{2}(\omega )}$

2. Kakalian

${\displaystyle x_{1}(t)\cdot x_{2}(t)\,}$	${\displaystyle {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{\sqrt {2\pi }}}\cdot (X_{1}*X_{2})(\omega )\,}$	(konvensasi normalisasi uniter)
	${\displaystyle {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{2\pi }}\cdot (X_{1}*X_{2})(\omega )\,}$	(konvensi non-uniter)
	${\displaystyle {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad (F*G)(f)}$	(frékuénsi biasa)

3. Modulasi:

${\displaystyle {\begin{aligned}x(t)\cdot \cos \omega _{0}t&\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{2}}[X(\omega +\omega _{0})+X(\omega -\omega _{0})],\qquad \omega _{0}\in \mathbb {R} \\f(t)\cdot \sin \omega _{0}t&\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {j}{2}}[X(\omega +\omega _{0})-X(\omega -\omega _{0})]\\x(t)\cdot e^{j\omega _{0}t}&\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad X(\omega -\omega _{0})\end{aligned}}\,}$

4. Géséran waktu

${\displaystyle x(t-t_{0})\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad e^{-j\omega t_{0}}\cdot X(\omega )}$

5. Géséran frékuénsi:

${\displaystyle x(t)\cdot e^{j\omega _{o}t}\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad X(\omega -\omega _{o})}$

6. Skala:

${\displaystyle x(at)\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{|a|}}X{\biggl (}{\frac {\omega }{a}}{\biggr )},\qquad a\in \mathbb {R} ,a\neq 0}$

7. Lawan / kabalikan waktu:

${\displaystyle x(-t)\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad X(-\omega )}$

8. Dualitas:

${\displaystyle X(t)\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad 2\pi x(-\omega )}$

9. Diferensiasi waktu:

${\displaystyle x^{'}(t)={\frac {dx(t)}{dt}}x(t)\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad j\omega X(\omega )}$

10. Diferensiasi frékuénsi:

${\displaystyle (-jt)x(t)\quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad X^{'}(\omega )={\frac {dX(\omega )}{dw}}}$

11. Integrasi:

${\displaystyle \int _{-\infty }^{t}x(\tau )\,d\tau \quad {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{j\omega }}X(\omega )+\pi X(0)\cdot \delta (\omega ),\,}$

Transformasi Fourier tina sawatara sinyal nu mangfaat

No.	Fungsi waktu	Transfirmasi Fourier (doméin Frékuénsi)
1.	${\displaystyle \delta (t)}$	1
2.	${\displaystyle \delta (t-t_{o})}$	${\displaystyle e^{-j\omega t_{o}}}$
3.	1	${\displaystyle 2\pi \delta (\omega )}$
4.	${\displaystyle e^{j\omega t_{o}}}$	${\displaystyle 2\pi \delta (\omega -\omega _{o})}$
5.	${\displaystyle \cos(\omega _{o}t)\,}$	${\displaystyle \pi \delta (\omega -\omega _{o})+\pi \delta (\omega +\omega _{o})}$
6.	${\displaystyle \sin(\omega _{o}t)\,}$	${\displaystyle -j\pi \delta (\omega -\omega _{o})+j\pi \delta (\omega +\omega _{o})}$
7.	${\displaystyle u(t)}$	${\displaystyle \pi \delta (\omega )+{\frac {1}{j\omega }}\,}$
8.	${\displaystyle e^{-at}u(t)}$	${\displaystyle {\frac {1}{j\omega +a}}\,}$ pikeun a>0
9.	${\displaystyle e^{-a|t|}}$	${\displaystyle {\frac {2a}{\omega ^{2}+a^{2}}}\,}$ pikeun a>0

Tempo ogé

• Dérét Fourier
• Transformasi Fourier gancang (Fast Fourier transform, FFT)
• Transformasi Laplace
• Transformasi Fourier diskrit
• Transformasi Fourier fraksional
• Transformasi kanonik liniér
• Transformasi sinus Fourier
• Transformasi Fourier laun (Short-time Fourier transform)
• Pamrosésan sinyal analog

Rujukan

• Всё о Mathcad Archived 2019-10-20 di Wayback Machine Citakan:Ref-ru
• Fourier Transforms from eFunda - includes tables
• Dym & McKéan, Fourier Series and Integrals. (For réaders with a background in mathematical analysis.)
• K. Yosida, Functional Analysis, Springer-Verlag, 1968. ISBN 3-540-58654-7
• L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, 1976. (Somewhat terse.)
• A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
• R. G. Wilson, "Fourier Series and Optical Transform Techniques in Contemporary Optics", Wiley, 1995. ISBN 0471303577
• R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed., Boston, McGraw Hill, 2000.

Tumbu kaluar

• Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.
• (en) Eric W. Weisstein, Fourier Transform di MathWorld.
• Fourier Transform Module by John H. Mathews
• Extending Laplace & Fourier Transforms by Dr. Shervin Erfani