Skip to content Skip to navigation Skip to collection information

OpenStax-CNX

You are here: Home » Content » Signals, Systems, and Society » Evaluation of Convolution Integrals

Navigation

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.
 

Evaluation of Convolution Integrals

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

The key to evaluating a convolution integral such as

x ( t ) * h ( t ) = - x ( τ ) h ( t - τ ) d τ x ( t ) * h ( t ) = - x ( τ ) h ( t - τ ) d τ
(1)

is to realize that as far as the integral is concerned, the variable tt is a constant and the integral is over the variable ττ. Therefore, for each tt, we are finding the area of the product x(τ)h(t-τ)x(τ)h(t-τ). Let's look at an example that illustrates how this works.

Example 3.1 Find the convolution of x(t)=u(t)x(t)=u(t) and h(t)=e-tu(t)h(t)=e-tu(t). The convolution integral is given by

h ( t ) * x ( t ) = - e - τ u ( τ ) u ( t - τ ) d τ h ( t ) * x ( t ) = - e - τ u ( τ ) u ( t - τ ) d τ
(2)

Figure 1 shows the graph of e-τu(τ)e-τu(τ), e-tu(t)e-tu(t), and their product. From the graph of the product, it is easy to see the the convolution integral becomes

0 t e - τ d τ = 1 - e - t , t 0 0 , t < 0 0 t e - τ d τ = 1 - e - t , t 0 0 , t < 0
(3)
Figure 1: Graphs of signals used in Example (Reference).
(a)
Figure 1(a) (ch3_exp.png)
(b)
Figure 1(b) (ch3_u.png)
(c)
Figure 1(c) (ch3_prod.png)

Signals which can be expressed in functional form should be convolved as in the above example. Other signals may not have an easy functional representation but rather may be piece-wise linear. In order to convolve such signals, one must evaluate the convolution integral over different intervals on the tt-axis so that each distinct interval corresponds to a different expression for x(t)*h(t)x(t)*h(t). The following example illustrates this:

Example 3.2 Suppose we attempt to convolve the unit step function x(t)=u(t)x(t)=u(t) with the trapezoidal function

h ( t ) = t , 0 t < 1 1 , 1 t < 2 0 , elsewhere h ( t ) = t , 0 t < 1 1 , 1 t < 2 0 , elsewhere
(4)

From Figure 2, it can be seen that on the interval 0t<10t<1, the product x(t-τ)h(τ)x(t-τ)h(τ) is an equilateral triangle with area t2/2t2/2. On the interval 1t<21t<2, the area of x(t-τ)h(τ)x(t-τ)h(τ) is t-1/2t-1/2. This latter area results by adding the area of an equilateral triangle having a base of 1, and the area of a rectangle having a base of t-1t-1 and a height of 1. For all values of tt greater than 2, the convolution is 1.5 since x(t-τ)h(τ)=h(τ)x(t-τ)h(τ)=h(τ) and h(τ)h(τ) is a trapezoid having an area of 1.5. Finally, for t<0t<0, the convolution is zero since x(t-τ)h(τ)=0x(t-τ)h(τ)=0.

Figure 2: Graphs of signals used in Example (Reference).
(a)
Figure 2(a) (ch3_trap.png)
(b)
Figure 2(b) (ch3_u1.png)

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 | 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