Skip to content Skip to navigation

OpenStax-CNX

You are here: Home » Content » m08 - Properties of the Laplace Transform

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

m08 - Properties of the Laplace Transform

Module by: C. Sidney Burrus. E-mail the author

Summary: The Laplace Transform has properties similar to those of the Fourier Transform and Fourier series.

Properties of the Laplace Transform

Many of the properties of the Laplace transform are similar to those for Fourier transform [1][2], however, the basis functions for the Laplace transform are not orthogonal. Some of the more important ones are:

  1. Linear: { x + y } = { x } + { y } { x + y } = { x } + { y }
  2. Convolution: If y ( t ) = h ( t ) * x ( t ) = h ( t τ ) x ( τ ) τ y ( t ) = h ( t ) * x ( t ) = h ( t τ ) x ( τ ) τ then { h ( t ) * x ( t ) } = { h ( t ) } { x ( t ) } { h ( t ) * x ( t ) } = { h ( t ) } { x ( t ) }
  3. Derivative: { x t } = s { x ( t ) } { x t } = s { x ( t ) }
  4. Derivative (ULT): { x t } = s { x ( t ) } x ( 0 ) { x t } = s { x ( t ) } x ( 0 )
  5. Integral: { x ( t ) t } = 1 s { x ( t ) } { x ( t ) t } = 1 s { x ( t ) }
  6. Shift: { x ( t T ) } = C ( k ) e T s { x ( t T ) } = C ( k ) e T s
  7. Modulate: { x ( t ) e j ω 0 t } = X ( s j ω 0 ) { x ( t ) e j ω 0 t } = X ( s j ω 0 )

References

  1. R. N. Bracewell. (1985). The Fourier Transform and Its Applications. (Third). New York: McGraw-Hill.
  2. A. Papoulis. (1962). The Fourier Integral and Its Applications. McGraw-Hill.

Content actions

Download module as:

Add 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