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# Konvolusi

Dina matematika sarta, hususna, analisis fungsional, konvolusi nyaéta operator matematis anu ngajadikeun dua fungsi x1 jeung x2 jadi fungsi katilu nu dianggap salaku vérsi modifikasi tina fungsi-fungsi asal. Konvolusi mangrupa hiji alat matematika dina élmu statistik, citra, pamrosésan sinyal, sarta persamaan diférential.

## Définisi

Konvolusi tina dua sinyal ${\displaystyle x_{1}(t)}$ jeung ${\displaystyle x_{2}(t)}$, nu dilambangkeun ku ${\displaystyle x_{1}(t)*x_{2}(t)}$ nyaéta hiji sinyal anyar x(t) anu didéfinisikeun ku:

${\displaystyle x(t)=x_{1}(t)*x_{2}(t)=(x_{1}*x_{2})(t)=\int _{-\infty }^{\infty }x_{1}(\tau )x_{2}(t-\tau )\,d\tau .}$

## Sifat konvolusi

${\displaystyle x_{1}(t)*x_{2}(t)=x_{2}(t)*x_{1}(t)\,}$
${\displaystyle x_{1}(t)*[x_{2}(t)*x_{3}(t)]=[x_{1}(t)*x_{2}(t)]*x_{3}(t)\,}$
${\displaystyle x_{1}(t)*[x_{2}(t)+x_{3}(t)]=[x_{1}(t)*x_{2}(t)]+[x_{1}(t)*x_{3}(t)]\,}$

## Konvolusi jeung fungsi ${\displaystyle \delta }$

${\displaystyle x_{1}(t)*\delta (t)=\delta (t)*x_{1}(t)=x_{1}(t)\,}$
${\displaystyle x_{1}(t)*\delta (t-t_{o})=\delta (t-t_{o})*x_{1}(t-t_{o})=x_{1}(t-t_{o})\,}$

## Téoréma konvolusi

Lamun ${\displaystyle {\mathcal {F}}\{x_{1}(t)\}\,}$ atawa ${\displaystyle X_{1}(\omega )}$ ngalambangkeun transformasi Fourier tina fungsi ${\displaystyle x_{1}(t)}$, ${\displaystyle {\mathcal {F}}\{x_{2}(t)\}\,}$ atawa ${\displaystyle X_{2}(\omega )}$ ngalambangkeun transformasi Fourier tina fungsi ${\displaystyle x_{2}(t)}$, sarta ${\displaystyle k}$ hiji konstanta, mangka:

${\displaystyle {\mathcal {F}}\{x_{1}(t)*x_{2}(t)\}=k\cdot {\mathcal {F}}\{x_{1}(t)\}\cdot {\mathcal {F}}\{x_{2}(t)\}=X_{1}(\omega )\cdot X_{2}(\omega )}$

sarta:

${\displaystyle x_{1}(t)*x_{2}(t){\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad X_{1}(\omega )\cdot X_{2}(\omega )\,}$

${\displaystyle x_{1}(t)\cdot x_{2}(t){\stackrel {\mathcal {F}}{\Longleftrightarrow }}\quad {\frac {1}{2\pi }}\cdot X_{1}(\omega )*X_{2}(\omega )\,}$

## Tumbu kaluar

Tempo convolution dina Wikikamus, kamus bébas.

## Rujukan

1. Hsu, Hwei P., Schaum's Outline of Théory and Problems of Analog and Digital Communications, McGraw Hill, 1993