Transformasi Fourier

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.

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

Définisi

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

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

Kabalikan transformasi Fourier $X(\omega )$ dilambangkeun ku Peta ''parse'' gagal (Kasalahan rumpaka): {\displaystyle \mathcal {F^’}\{} } sarta didéfiniskieun kieu:

Peta ''parse'' gagal (Kasalahan rumpaka): {\displaystyle x(t) = \mathcal {F^’}\{X(\omega)\} = \frac{1}{2\pi} \int _{-\infty}^{\infty} X(\omega)\ e^{ j\omega t}\,d\omega }

$x(t)jeungX(\omega )$ disebut pasangan transformasi Fourier.

Sifat Transformasi Fourier

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

1. Liniéritas (superposisi):

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

2. Kakalian

$x_{1}(t)\cdot x_{2}(t)\,$	${\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{\sqrt {2\pi }}}\cdot (X_{1}*X_{2})(\omega )\,$	(konvensasi normalisasi uniter)
	${\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{2\pi }}\cdot (X_{1}*X_{2})(\omega )\,$	(konvensi non-uniter)
	Peta ''parse'' gagal (fungsi teu kanyahoan): {\displaystyle \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad (F*G)(\f) \,}	(frekuensi biasa)

3. Modulasi:

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

4. Géséran waktu

Peta ''parse'' gagal (Kasalahan rumpaka): {\displaystyle x(t – t_o) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad X(\omega)\ e^{-j \omega t_o} }

5. Géséran frékuénsi:

Peta ''parse'' gagal (Kasalahan rumpaka): {\displaystyle x(t) \ e^{j \omega_o t} \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad X(\omega – \omega_o) }

6. Skala:

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

7. Lawan / kabalikan waktu:

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

8. Dualitas:

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

9. Diferensiasi waktu:

Peta ''parse'' gagal (Mun bisa MathML (uji coba): Respons tak sah ("Math extension cannot connect to Restbase.") dari peladen "http://localhost:6011/su.wikipedia.org/v1/":): {\displaystyle x^’ (t) = \frac{d x(t)}{dt}\ x(t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad j\omega X(\omega) }

10. Diferensiasi frékuénsi:

Peta ''parse'' gagal (Kasalahan rumpaka): {\displaystyle (-jt) x(t) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad X^’ (\omega) = \frac{d X(\omega)}{dw} }

11. Integrasi:

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

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

• 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 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-10: 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

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