# 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