SIGNAL PROCESSING

# Convolution

 Date Published: June 5, 2018 Last Modified: December 24, 2018

## Overview

Convolution is a mathematic operation that can be performed on two functions, which produces a third output function which is a “blend” of the two inputs.

Convolution can be thought of as a measure of the amount of overlap of one function as it is shifted over the other function

## Formal Definition

$$f \ast g = \int_{-\infty}^{\infty} f(\tau)\ g(t - \tau) d \tau$$

## Mathematical Properties

Convolution is commutative:

$$f \ast g = g \ast f$$

Convolution is associative:

$$(f \ast g) \ast h = f \ast (g \ast h)$$

Convolution is distributive:

$$f \ast (g + h) = f \ast g + f \ast h$$

These other properties also hold true:

$$a (f \ast g) = (af) \ast g$$

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