Convolution

Overview

This is a operation that does basic composition of functions like $+- \frac{d}{dx}$

Convolution of $f(x)$ with $g(x)$ denoted as:

$$(f*g)(s) = \int_{-\infty}^{\infty}f(x)\cdot g(s-x) \, dx$$

• $g(s-x)$ is known as the kernel
  • This is flipped horizontally ($-x$) and shifted by $s$

You have a function $f(x)$, you define a kernel $g(x)$ and then you flip that kernal, shift it by variable amount and plot the sum of underlying graph.

Below is an interactive tutorial about how convolution works - https://www.desmos.com/calculator/0jjwruqbjo

Properties of Convolution

1. Commutative $f*g = g*f$
2. Associative
   $f*(g*h)=(f*g)*h$
3. Distributive
   $f*(g+h)=(f*g)+(f*h)$
4. Identity $f*\delta=f$ where $\delta$ is the Dirac delta function

Special Properties

Laplace and Fourier transforms of a Convolution = Multiplication of those transforms separately.

$$\mathcal{L}\{f*g\} = \mathcal{L}\{f\} \cdot \mathcal{L}\{g\}$$ $$\mathcal{F}\{f*g\} = \mathcal{F}\{f\} \cdot \mathcal{F}\{g\}$$