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 mangrupakeun 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:

$x(t) = x_1 (t) * x_2 (t) = (x_1 * x_2 )(t) = \int_{-\infty}^{\infty} x_1(\tau) x_2(t - \tau)\, d\tau.$

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

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

sarta:

$x_1 (t) * x_2 (t) \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad X_1(\omega)\cdot X_2(\omega) \,$

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

Tempo ogé

• Toeplitz matrix (convolutions can be considered a Toeplitz matrix operation where each 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

Tempo convolution dina Wiktionary, kamus bébas.

• http://www.nitte.ac.in/downloads/Conv-LTI.pdf
• Convolution, 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
• Convolution Kernel Mask Operation Interactive tutorial
• Convolution at MathWorld

Rujukan

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