Convolution (Signal Processing)

Definition 1: Convolution

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

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

given that the integral converges.

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 exist $\alpha ,\beta \in \mathbb{R}$ such that the integrals

${\int }_{-\infty }^{\infty }|f(t)|e^{-\alpha t}dt\text{and}{\int }_{-\infty }^{\infty }|g(t)|e^{-\beta t}dt$ converge then,

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

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:

${\mathcal{L}}^{-1}\{F(s)G(s)\}=(f\ast g)(t)$

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)!!

Related

• Laplace Transform