# OpenStax_CNX

You are here: Home » Content » Properties of Convolution Integrals

### Recently Viewed

This feature requires Javascript to be enabled.

# Properties of Convolution Integrals

Module by: Carlos E. Davila. E-mail the author

We list several important properties and their proofs.

1. Commutative Property:
x(t)*h(t)=h(t)*x(t)x(t)*h(t)=h(t)*x(t)
(1)
x(t)*h(t)=-x(τ)h(t-τ)dτx(t)*h(t)=-x(τ)h(t-τ)dτ
(2)
and make the substitution γ=t-τγ=t-τ. It follows that
x(t)*h(t)=-x(t-γ)h(γ)dγdτ=h(t)*x(t)x(t)*h(t)=-x(t-γ)h(γ)dγdτ=h(t)*x(t)
(3)
2. Associative Property:
x(t)*h1(t)*h2(t)=x(t)*h1(t)*h2(t)x(t)*h1(t)*h2(t)=x(t)*h1(t)*h2(t)
(4)
To prove this property we begin with an expression for the left-hand side of Equation 4
-x(τ)h1(t-τ)dτ*h2(t)-x(τ)h1(t-τ)dτ*h2(t)
(5)
where we have expressed x(t)*h1(t)x(t)*h1(t) as a convolution integral. Expanding the second convolution gives
--x(τ)h1(γ-τ)dτh2(t-γ)dγ--x(τ)h1(γ-τ)dτh2(t-γ)dγ
(6)
Reversing the order of integration gives
-x(τ)-h1(γ-τ)h2(t-γ)dγdτ-x(τ)-h1(γ-τ)h2(t-γ)dγdτ
(7)
Using the variable substitution φ=γ-τφ=γ-τ and integrating over φφ in the inner integral gives the final result:
-x(τ)-h1(φ)h2(t-τ-φ)dγdτ-x(τ)-h1(φ)h2(t-τ-φ)dγdτ
(8)
where the inner integral is recognized as h1(t)*h2(t)h1(t)*h2(t) evaluated at t=t-τt=t-τ, which is required for the convolution with x(t)x(t).
3. Distributive Property:
x(t)*h1(t)+h2(t)=x(t)*h1(t)+x(t)*h2(t)x(t)*h1(t)+h2(t)=x(t)*h1(t)+x(t)*h2(t)
(9)
This property is easily proven from the definition of the convolution integral.
4. Time-Shift Property: If y(t)=x(t)*h(t)y(t)=x(t)*h(t) then x(t-t0)*h(t)=y(t-t0)x(t-t0)*h(t)=y(t-t0) Again, the proof is trivial.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks