Konvolíció

Konvolíció, also known as convolution, is a mathematical operation on two functions that produces a third function expressing how the shape of one is modified by the other. It is a fundamental concept in various fields such as signal processing, image processing, and probability theory.

In mathematics, the convolution of two functions f and g is denoted by (f * g) and is defined as:

(f * g)(t) = ∫ from -∞ to ∞ f(τ) g(t - τ) dτ

This integral represents the sum of the products of the two functions, where one function is shifted and scaled by the other. The convolution operation is commutative, meaning (f * g) = (g * f).

In signal processing, convolution is used to analyze the effect of a system on an input signal. The output signal is the convolution of the input signal and the system's impulse response. In image processing, convolution is used for various tasks such as blurring, sharpening, and edge detection.

In probability theory, the convolution of two independent random variables X and Y is the distribution of their sum. If X and Y are independent and have probability density functions f and g, respectively, then the probability density function of X + Y is (f * g).

The convolution operation has several important properties, including linearity, associativity, and commutativity. It is also closely related to the Fourier transform, which is a powerful tool for analyzing the frequency content of signals.