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.

Gambaran visual ngeunaan konvolusi.

Définisi

Konvolusi tina dua sinyal $x_1(t)$ jeung $x_2(t)$, nu dilambangkeun ku $x_1(t)*x_2(t)$ nyaéta hiji sinyal anyar x(t) anu didéfinisikeun ku:

$x(t)=x_1(t)*x_2(t)=(x_1*x_2)(t)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{x}_1(\tau )x_2(t-\tau )d\tau .$

Sifat konvolusi

$x_1(t)*x_2(t)=x_2(t)*x_1(t)$

$x_1(t)*[x_2(t)*x_3(t)]=[x_1(t)*x_2(t)]*x_3(t)$

$x_1(t)*[x_2(t)+x_3(t)]=[x_1(t)*x_2(t)]+[x_1(t)*x_3(t)]$

Konvolusi jeung fungsi $\delta$

$x_1(t)*\delta (t)=\delta (t)*x_1(t)=x_1(t)$

$x_1(t)*\delta (t-t_o)=\delta (t-t_o)*x_1(t-t_o)=x_1(t-t_o)$

Téoréma konvolusi

Lamun $ℱ\{x_1(t)\}$ atawa $X_1(\omega )$ ngalambangkeun transformasi Fourier tina fungsi $x_1(t)$, $ℱ\{x_2(t)\}$ atawa $X_2(\omega )$ ngalambangkeun transformasi Fourier tina fungsi $x_2(t)$, sarta $k$ hiji konstanta, mangka:

$ℱ\{x_1(t)*x_2(t)\}=k\cdot ℱ\{x_1(t)\}\cdot ℱ\{x_2(t)\}=X_1(\omega )\cdot X_2(\omega )$

sarta:

$x_1(t)*x_2(t)\stackrel{ℱ}{⟺}{X}_1(\omega )\cdot X_2(\omega )$

$x_1(t)\cdot x_2(t)\stackrel{ℱ}{⟺}\frac{1}{2\pi }\cdot X_1(\omega )*X_2(\omega )$

Tempo ogé

• Toeplitz matrix (convolutions can be considered a Toeplitz matrix operation where éach row is a shifted copy of the convolution kernel)
  • Circulant matrix
• Cross-correlation
• Deconvolution
• Dirichlet convolution
• Titchmarsh convolution theorem
• Convolution power
• Analog signal processing
• List of convolutions of probability distributions

Tumbu kaluar

• http://www.nitte.ac.in/downloads/Conv-LTI.pdf Citakan:Webarchive
• Convolution Citakan:Webarchive, on The Data Analysis BriefBook
• http://www.jhu.edu/~signals/convolve/index.html Visual convolution Java Applet.
• http://www.jhu.edu/~signals/discreteconv2/index.html Visual convolution Java Applet for Discrete Time functions.
• Lectures on Image Processing: A collection of 18 lectures in pdf format from Vanderbilt University. Lecture 7 is on 2-D convolution., by Alan Peters.
  • http://www.vuse.vanderbilt.edu/~rap2/EECE253/EECE253_01_Intro.pdf Citakan:Webarchive
• Convolution Kernel Mask Operation Interactive tutorial
• Convolution at MathWorld

Rujukan

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