Transformasi Fourier

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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.

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 Fourier

Définisi[édit | sunting sumber]

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  \mathcal {F^'} sarta didéfiniskieun kieu:

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.

dimana  x(t) jeung X(\omega) disebut pasangan transformasi Fourier.

Sifat Transformasi Fourier[édit | sunting sumber]

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)
    \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad 
(F*G)(f)     (frekuensi biasa)

3. Modulasi:


\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}
\,

4. Géséran waktu

x(t-t_0) \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad e^{-j\omega t_0}\cdot X(\omega)

5. Géséran frékuénsi:

 x(t)\cdot 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 \ne 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:

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

10. Diferensiasi frékuénsi:

 (-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),
\,

Transformasi Fourier tina sawatara sinyal nu mangfaat[édit | sunting sumber]

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

Tempo ogé[édit | sunting sumber]

Rujukan[édit | sunting sumber]

  • Всё о Mathcad Citakan:Ref-ru
  • 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[édit | sunting sumber]