Skip to content Skip to navigation Skip to collection information

Connexions

You are here: Home » Content » Signals and Systems » Laplace Transform

Navigation

Table of Contents

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.

Also in these lenses

  • Lens for Engineering

    This module and collection are 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 collection is included inLens: richb's DSP resources
    By: Richard Baraniuk

    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.
Download
x

Download collection as:

  • PDF
  • EPUB (what's this?)

    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 "(what's this?)" link.

  • More downloads ...

Download module as:

  • PDF
  • EPUB (what's this?)

    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 "(what's this?)" link.

  • More downloads ...
Reuse / Edit
x

Collection:

Module:

Add to a lens
x

Add collection to:

Add module to:

Add to Favorites
x

Add collection to:

Add module to:

 

Laplace Transform

Module by: Richard Baraniuk, Dan Calderon. E-mail the authors

Summary: Describes Laplace transforms.

Introduction

The Laplace transform is a generalization of the Continuous-Time Fourier Transform. It is used because the CTFT does not converge/exist for many important signals, and yet it does for the Laplace-transform (e.g., signals with infinite l 2 l 2 norm). It is also used because it is notationally cleaner than the CTFT. However, instead of using complex exponentials of the form ej⁢ω⁢t ω t , with purely imaginary parameters, the Laplace transform uses the more general, es⁢t s t , where s=σ+j⁢ω s σ ω is complex, to analyze signals in terms of exponentially weighted sinusoids.

The Laplace Transform

Bilateral Laplace Transform Pair

Although Laplace transforms are rarely solved in practice using integration (tables and computers (e.g. Matlab) are much more common), we will provide the bilateral Laplace transform pair here for purposes of discussion and derivation. These define the forward and inverse Laplace transformations. Notice the similarities between the forward and inverse transforms. This will give rise to many of the same symmetries found in Fourier analysis.

Laplace Transform

F⁢s=∫−∞∞f⁢t⁢e−(s⁢t)d t F s t f t s t
(1)

Inverse Laplace Transform

f⁢t=12⁢π⁢j⁢∫c−j⁢∞c+j⁢∞F⁢s⁢es⁢td s f t 1 2 s c c F s s t
(2)

Note:

We have defined the bilateral Laplace transform. There is also a unilateral Laplace transform ,
F⁢s=∫0∞f⁢t⁢e−(s⁢t)d t F s t 0 f t s t
(3)
which is useful for solving the difference equations with nonzero initial conditions. This is similar to the unilateral Z Transform in Discrete time.

Relation between Laplace and CTFT

Taking a look at the equations describing the Z-Transform and the Discrete-Time Fourier Transform:

Continuous-Time Fourier Transform

ℱ⁢Ω=∫−∞∞f⁢t⁢e−(j⁢Ω⁢t)d t ℱ Ω t f t Ω t
(4)

Laplace Transform

F⁢s=∫−∞∞f⁢t⁢e−(s⁢t)d t F s t f t s t
(5)
We can see many similarities; first, that :
ℱ⁢Ω=F⁢s ℱ Ω Fs
(6)
for all Ω=s Ω s

Note:

the CTFT is a complex-valued function of a real-valued variable ω ω (and 2 π periodic). The Z-transform is a complex-valued function of a complex valued variable z.
Figure 1
Plots
Plots (splanefigure1.png)

Visualizing the Laplace Transform

With the Fourier transform, we had a complex-valued function of a purely imaginary variable, F⁢j⁢ω F ω . This was something we could envision with two 2-dimensional plots (real and imaginary parts or magnitude and phase). However, with Laplace, we have a complex-valued function of a complex variable. In order to examine the magnitude and phase or real and imaginary parts of this function, we must examine 3-dimensional surface plots of each component.

Figure 2: Real and imaginary parts of H⁢s H s are now each 3-dimensional surfaces.
real and imaginary sample plots
(a) The Real part of H⁢s H s (b) The Imaginary part of H⁢s H s
Figure 2(a) (laplace1.png)Figure 2(b) (laplace2.png)
Figure 3: Magnitude and phase of H⁢s H s are also each 3-dimensional surfaces. This representation is more common than real and imaginary parts.
magnitude and phase sample plots
(a) The Magnitude of H⁢s H s (b) The Phase of H⁢s H s
Figure 3(a) (laplace3.png)Figure 3(b) (laplace4.png)

While these are legitimate ways of looking at a signal in the Laplace domain, it is quite difficult to draw and/or analyze. For this reason, a simpler method has been developed. Although it will not be discussed in detail here, the method of Poles and Zeros is much easier to understand and is the way both the Laplace transform and its discrete-time counterpart the Z-transform are represented graphically.

Using a Computer to find the Laplace Transform

Using a computer to find Laplace transforms is relatively painless. Matlab has two functions, laplace and ilaplace, that are both part of the symbolic toolbox, and will find the Laplace and inverse Laplace transforms respectively. This method is generally preferred for more complicated functions. Simpler and more contrived functions are usually found easily enough by using tables.

Laplace Transform Definition Demonstration

Figure 4: Interact (when online) with a Mathematica CDF demonstrating the Laplace Transform. To Download, right-click and save target as .cdf.
LaplaceTransformDemo

Interactive Demonstrations

Figure 5: See the attached video on the basics of the Unilateral Laplace Transform from Khan Academy
Khan Lecture on Laplace

Conclusion

The laplace transform proves a useful, more general form of the Continuous Time Fourier Transform. It applies equally well to describing systems as well as signals using the eigenfunction method, and to describing a larger class of signals better described using the pole-zero method.

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

Reuse / Edit:

Reuse or edit collection (?)

Check out and edit

If you have permission to edit this content, using the "Reuse / Edit" action will allow you to check the content out into your Personal Workspace or a shared Workgroup and then make your edits.

Derive a copy

If you don't have permission to edit the content, you can still use "Reuse / Edit" to adapt the content by creating a derived copy of it and then editing and publishing the copy.

| Reuse or edit module (?)

Check out and edit

If you have permission to edit this content, using the "Reuse / Edit" action will allow you to check the content out into your Personal Workspace or a shared Workgroup and then make your edits.

Derive a copy

If you don't have permission to edit the content, you can still use "Reuse / Edit" to adapt the content by creating a derived copy of it and then editing and publishing the copy.