Convolution (Signal Processing)

Watch 3Blue1Brown video: But what is convolution?

Definition 1: Convolution

The convolution of the functions $f(t)$ and $g(t)$, denoted by $(f\star g)(t)$ is the integral

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

given that the integral converges.

Define the idea of an integral transform.

Definition 2: Integral Transform

An integral transform is a functional (function on a set of functions) that can be written in the form

$\mathcal{T}\{f(t)\}={\int }_{-\infty }^{\infty }f(t)K(t,s)dt$.

The function $K(t,u)$ is called the kernel of the integral transform.

Convolution Properties

Theorem 1: Convolution Properties

A. The convolution operator is commutative $(f\ast g)(t)=(g\ast f)(t)$ B. If $f(t)$ and $g(t)$ are one-sided functions (i.e. $f(t)=g(t)=0$ for $t<0$) then

$(f\ast g)(t)={\int }_0^{t}f(\tau )g(t-\tau )d\tau$

and hence the convolution is also one sided.

Theorem 2: Convolution Theorem

If there exists $\alpha ,\beta \in \mathbb{R}$ such that the integrals

${\int }_{-\infty }^{\infty }|f(t)|e^{-\alpha t}dt$

and

${\int }_{-\infty }^{\infty }|g(t)|e^{-\beta t}dt$

converge then,

$\mathcal{L}\{(f\ast g)(t)\}=F(s)G(s)$