Skip to content Skip to navigation Skip to collection information

Connexions

You are here: Home » Content » Information and Signal Theory » Properties of Continuous Time Convolution

Navigation

Lenses

What is a lens?

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? tag icon

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

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • OrangeGrove display tagshide tags

    This module is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange GroveAs a part of collection: "Signals and Systems"

    Click the "OrangeGrove" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Rice Digital Scholarship display tagshide tags

    This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Signals and Systems"

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • Lens for Engineering

    This module is included inLens: Lens for Engineering
    By: Sidney Burrus

    Click the "Lens for Engineering" link to see all content selected in this lens.

  • richb's DSP display tagshide tags

    This module is included inLens: richb's DSP resources
    By: Richard BaraniukAs a part of collection: "Signals and Systems"

    Comments:

    "My introduction to signal processing course at Rice University."

    Click the "richb's DSP" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Properties of Continuous Time Convolution

Module by: Melissa Selik, Richard Baraniuk, Stephen Kruzick. E-mail the authors

Summary: This module discusses the properties of continuous time convolution.

Introduction

We have already shown the important role that continuous time convolution plays in signal processing. This section provides discussion and proof of some of the important properties of continuous time convolution. Analogous properties can be shown for continuous time circular convolution with trivial modification of the proofs provided except where explicitly noted otherwise.

Continuous Time Convolution Properties

Associativity

The operation of convolution is associative. That is, for all continuous time signals f1,f2,f3f1,f2,f3 the following relationship holds.

f 1 * ( f 2 * f 3 ) = ( f 1 * f 2 ) * f 3 f 1 * ( f 2 * f 3 ) = ( f 1 * f 2 ) * f 3
(1)

In order to show this, note that

( f 1 * ( f 2 * f 3 ) ) ( t ) = ∫ - ∞ ∞ ∫ - ∞ ∞ f 1 ( τ 1 ) f 2 ( τ 2 ) f 3 ( ( t - τ 1 ) - τ 2 ) d τ 2 d τ 1 = ∫ - ∞ ∞ ∫ - ∞ ∞ f 1 ( τ 1 ) f 2 ( ( τ 1 + τ 2 ) - τ 1 ) f 3 ( t - ( τ 1 + τ 2 ) ) d τ 2 d τ 1 = ∫ - ∞ ∞ ∫ - ∞ ∞ f 1 ( τ 1 ) f 2 ( τ 3 - τ 1 ) f 3 ( t - τ 3 ) d τ 1 d τ 3 = ( ( f 1 * f 2 ) * f 3 ) ( t ) ( f 1 * ( f 2 * f 3 ) ) ( t ) = ∫ - ∞ ∞ ∫ - ∞ ∞ f 1 ( τ 1 ) f 2 ( τ 2 ) f 3 ( ( t - τ 1 ) - τ 2 ) d τ 2 d τ 1 = ∫ - ∞ ∞ ∫ - ∞ ∞ f 1 ( τ 1 ) f 2 ( ( τ 1 + τ 2 ) - τ 1 ) f 3 ( t - ( τ 1 + τ 2 ) ) d τ 2 d τ 1 = ∫ - ∞ ∞ ∫ - ∞ ∞ f 1 ( τ 1 ) f 2 ( τ 3 - τ 1 ) f 3 ( t - τ 3 ) d τ 1 d τ 3 = ( ( f 1 * f 2 ) * f 3 ) ( t )
(2)

proving the relationship as desired through the substitution τ3=τ1+τ2τ3=τ1+τ2.

Commutativity

The operation of convolution is commutative. That is, for all continuous time signals f1,f2f1,f2 the following relationship holds.

f 1 * f 2 = f 2 * f 1 f 1 * f 2 = f 2 * f 1
(3)

In order to show this, note that

( f 1 * f 2 ) ( t ) = ∫ - ∞ ∞ f 1 ( τ 1 ) f 2 ( t - τ 1 ) d τ 1 = ∫ - ∞ ∞ f 1 ( t - τ 2 ) f 2 ( τ 2 ) d τ 2 = ( f 2 * f 1 ) ( t ) ( f 1 * f 2 ) ( t ) = ∫ - ∞ ∞ f 1 ( τ 1 ) f 2 ( t - τ 1 ) d τ 1 = ∫ - ∞ ∞ f 1 ( t - τ 2 ) f 2 ( τ 2 ) d τ 2 = ( f 2 * f 1 ) ( t )
(4)

proving the relationship as desired through the substitution τ2=t-τ1τ2=t-τ1.

Distribitivity

The operation of convolution is distributive over the operation of addition. That is, for all continuous time signals f1,f2,f3f1,f2,f3 the following relationship holds.

f 1 * ( f 2 + f 3 ) = f 1 * f 2 + f 1 * f 3 f 1 * ( f 2 + f 3 ) = f 1 * f 2 + f 1 * f 3
(5)

In order to show this, note that

( f 1 * ( f 2 + f 3 ) ) ( t ) = ∫ - ∞ ∞ f 1 ( τ ) ( f 2 ( t - τ ) + f 3 ( t - τ ) ) d τ = ∫ - ∞ ∞ f 1 ( τ ) f 2 ( t - τ ) d τ + ∫ - ∞ ∞ f 1 ( τ ) f 3 ( t - τ ) d τ = ( f 1 * f 2 + f 1 * f 3 ) ( t ) ( f 1 * ( f 2 + f 3 ) ) ( t ) = ∫ - ∞ ∞ f 1 ( τ ) ( f 2 ( t - τ ) + f 3 ( t - τ ) ) d τ = ∫ - ∞ ∞ f 1 ( τ ) f 2 ( t - τ ) d τ + ∫ - ∞ ∞ f 1 ( τ ) f 3 ( t - τ ) d τ = ( f 1 * f 2 + f 1 * f 3 ) ( t )
(6)

proving the relationship as desired.

Multilinearity

The operation of convolution is linear in each of the two function variables. Additivity in each variable results from distributivity of convolution over addition. Homogenity of order one in each varible results from the fact that for all continuous time signals f1,f2f1,f2 and scalars aa the following relationship holds.

a ( f 1 * f 2 ) = ( a f 1 ) * f 2 = f 1 * ( a f 2 ) a ( f 1 * f 2 ) = ( a f 1 ) * f 2 = f 1 * ( a f 2 )
(7)

In order to show this, note that

( a ( f 1 * f 2 ) ) ( t ) = a ∫ - ∞ ∞ f 1 ( τ ) f 2 ( t - τ ) d τ = ∫ - ∞ ∞ ( a f 1 ( τ ) ) f 2 ( t - τ ) d τ = ( ( a f 1 ) * f 2 ) ( t ) = ∫ - ∞ ∞ f 1 ( τ ) ( a f 2 ( t - τ ) ) d τ = ( f 1 * ( a f 2 ) ) ( t ) ( a ( f 1 * f 2 ) ) ( t ) = a ∫ - ∞ ∞ f 1 ( τ ) f 2 ( t - τ ) d τ = ∫ - ∞ ∞ ( a f 1 ( τ ) ) f 2 ( t - τ ) d τ = ( ( a f 1 ) * f 2 ) ( t ) = ∫ - ∞ ∞ f 1 ( τ ) ( a f 2 ( t - τ ) ) d τ = ( f 1 * ( a f 2 ) ) ( t )
(8)

proving the relationship as desired.

Conjugation

The operation of convolution has the following property for all continuous time signals f1,f2f1,f2.

f 1 * f 2 ¯ = f 1 ¯ * f 2 ¯ f 1 * f 2 ¯ = f 1 ¯ * f 2 ¯
(9)

In order to show this, note that

( f 1 * f 2 ¯ ) ( t ) = ∫ - ∞ ∞ f 1 ( τ ) f 2 ( t - τ ) d τ ¯ = ∫ - ∞ ∞ f 1 ( τ ) f 2 ( t - τ ) ¯ d τ = ∫ - ∞ ∞ f 1 ¯ ( τ ) f 2 ¯ ( t - τ ) d τ = ( f 1 ¯ * f 2 ¯ ) ( t ) ( f 1 * f 2 ¯ ) ( t ) = ∫ - ∞ ∞ f 1 ( τ ) f 2 ( t - τ ) d τ ¯ = ∫ - ∞ ∞ f 1 ( τ ) f 2 ( t - τ ) ¯ d τ = ∫ - ∞ ∞ f 1 ¯ ( τ ) f 2 ¯ ( t - τ ) d τ = ( f 1 ¯ * f 2 ¯ ) ( t )
(10)

proving the relationship as desired.

Time Shift

The operation of convolution has the following property for all continuous time signals f1,f2f1,f2 where STST is the time shift operator.

S T ( f 1 * f 2 ) = ( S T f 1 ) * f 2 = f 1 * ( S T f 2 ) S T ( f 1 * f 2 ) = ( S T f 1 ) * f 2 = f 1 * ( S T f 2 )
(11)

In order to show this, note that

S T ( f 1 * f 2 ) ( t ) = ∫ - ∞ ∞ f 2 ( τ ) f 1 ( ( t - T ) - τ ) d τ = ∫ - ∞ ∞ f 2 ( τ ) S T f 1 ( t - τ ) d τ = ( ( S T f 1 ) * f 2 ) ( t ) = ∫ - ∞ ∞ f 1 ( τ ) f 2 ( ( t - T ) - τ ) d τ = ∫ - ∞ ∞ f 1 ( τ ) S T f 2 ( t - τ ) d τ = f 1 * ( S T f 2 ) ( t ) S T ( f 1 * f 2 ) ( t ) = ∫ - ∞ ∞ f 2 ( τ ) f 1 ( ( t - T ) - τ ) d τ = ∫ - ∞ ∞ f 2 ( τ ) S T f 1 ( t - τ ) d τ = ( ( S T f 1 ) * f 2 ) ( t ) = ∫ - ∞ ∞ f 1 ( τ ) f 2 ( ( t - T ) - τ ) d τ = ∫ - ∞ ∞ f 1 ( τ ) S T f 2 ( t - τ ) d τ = f 1 * ( S T f 2 ) ( t )
(12)

proving the relationship as desired.

Differentiation

The operation of convolution has the following property for all continuous time signals f1,f2f1,f2.

d d t ( f 1 * f 2 ) ( t ) = d f 1 d t * f 2 ( t ) = f 1 * d f 2 d t ( t ) d d t ( f 1 * f 2 ) ( t ) = d f 1 d t * f 2 ( t ) = f 1 * d f 2 d t ( t )
(13)

In order to show this, note that

d d t ( f 1 * f 2 ) ( t ) = ∫ - ∞ ∞ f 2 ( τ ) d d t f 1 ( t - τ ) d τ = d f 1 d t * f 2 ( t ) = ∫ - ∞ ∞ f 1 ( τ ) d d t f 2 ( t - τ ) d τ = f 1 * d f 2 d t ( t ) d d t ( f 1 * f 2 ) ( t ) = ∫ - ∞ ∞ f 2 ( τ ) d d t f 1 ( t - τ ) d τ = d f 1 d t * f 2 ( t ) = ∫ - ∞ ∞ f 1 ( τ ) d d t f 2 ( t - τ ) d τ = f 1 * d f 2 d t ( t )
(14)

proving the relationship as desired.

Impulse Convolution

The operation of convolution has the following property for all continuous time signals ff where δδ is the Dirac delta funciton.

f * δ = f f * δ = f
(15)

In order to show this, note that

( f * δ ) ( t ) = ∫ - ∞ ∞ f ( τ ) δ ( t - τ ) d τ = f ( t ) ∫ - ∞ ∞ δ ( t - τ ) d τ = f ( t ) ( f * δ ) ( t ) = ∫ - ∞ ∞ f ( τ ) δ ( t - τ ) d τ = f ( t ) ∫ - ∞ ∞ δ ( t - τ ) d τ = f ( t )
(16)

proving the relationship as desired.

Width

The operation of convolution has the following property for all continuous time signals f1,f2f1,f2 where Duration(f)Duration(f) gives the duration of a signal ff.

Duration ( f 1 * f 2 ) = Duration ( f 1 ) + Duration ( f 2 ) Duration ( f 1 * f 2 ) = Duration ( f 1 ) + Duration ( f 2 )
(17)

. In order to show this informally, note that (f1*f2)(t)(f1*f2)(t) is nonzero for all tt for which there is a ττ such that f1(τ)f2(t-τ)f1(τ)f2(t-τ) is nonzero. When viewing one function as reversed and sliding past the other, it is easy to see that such a ττ exists for all tt on an interval of length Duration(f1)+Duration(f2)Duration(f1)+Duration(f2). Note that this is not always true of circular convolution of finite length and periodic signals as there is then a maximum possible duration within a period.

Convolution Properties Summary

As can be seen the operation of continuous time convolution has several important properties that have been listed and proven in this module. With slight modifications to proofs, most of these also extend to continuous time circular convolution as well and the cases in which exceptions occur have been noted above. These identities will be useful to keep in mind as the reader continues to study signals and systems.

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

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

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

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

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

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? tag icon

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

Module to:

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? tag icon

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