# Convolution (Signal Processing)

Definition 1: Convolution

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

$(f∗g)(t)=∫_{−∞}f(τ)g(t−τ)dτ$

given that the integral converges.

### Convolution Properties

Theorem 1: Convolution Properties

A. The convolution operator is commutative $(f∗g)(t)=(g∗f)(t)$.

B. If $f(t)$ and $g(t)$ are one-sided functions (i.e. $f(t)=g(t)=0$ for $t<0$) then $(f∗g)(t)=∫_{0}f(τ)g(t−τ)dτ$ and hence the convolution is also one sided.

Theorem 2: Convolution Theorem

If there exist $α,β∈R$ such that the integrals

$∫_{−∞}∣f(t)∣e_{−αt}dtand∫_{−∞}∣g(t)∣e_{−βt}dt$ converge then,

$L{(f∗g)(t)}=F(s)G(s)$

This theorem states that the Laplace Transform of a convolution is the product of the transforms! A direct result of this allows us to “quickly” compute inverse Laplace transforms:

$L_{−1}{F(s)G(s)}=(f∗g)(t)$

Note: for real valued functions $f$ and $g$, the convolution is a real valued integral (i.e. not a contour integral in the complex plane)!!