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 Fourier

Daptar eusi

1 Définisi
2 Sifat Transformasi Fourier
3 Transformasi Fourier tina sawatara sinyal nu mangfaat
4 Tempo ogé
5 Rujukan
6 Tumbu kaluar

Définisi [édit]

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]

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:

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