Convolution (Signal Processing)

Definition 1: Convolution

The convolution of functions and , denoted by is the integral

given that the integral converges.

Convolution Properties

Theorem 1: Convolution Properties

A. The convolution operator is commutative .

B. If and are one-sided functions (i.e. for ) then and hence the convolution is also one sided.

Theorem 2: Convolution Theorem

If there exist such that the integrals

converge then,

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:

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