Skip to content Skip to navigation

Connexions

You are here: Home » Content » Parseval's Theorem

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

Parseval's Theorem

Module by: Don Johnson. E-mail the author

Summary: Information about Parseval's Theorem. The title says it all, really.

Properties of the Fourier transform and some useful transform pairs are provided in this table. Especially important among these properties is Parseval's Theorem, which states that power computed in either domain equals the power in the other.

∫−∞∞s2td t =∫−∞∞|S⁢f|2d f t s t 2 f S f 2
(1)
Of practical importance is the conjugate symmetry property: When s⁢t s t is real-valued, the spectrum at negative frequencies equals the complex conjugate of the spectrum at the corresponding positive frequencies. Consequently, we need only plot the positive frequency portion of the spectrum (we can easily determine the remainder of the spectrum).

Exercise 1

How many Fourier transform operations need to be applied to get the original signal back: ℱ⁢⋯⁢ℱ⁢s=s⁢t ℱ ⋯ ℱ s s t ?

Solution

ℱ⁢ℱ⁢ℱ⁢ℱ⁢s⁢t=s⁢t ℱ ℱ ℱ ℱ s t s t . We know that ℱ⁢S⁢f=∫−∞∞S⁢f⁢e−(i⁢2⁢π⁢f⁢t)d f =∫−∞∞S⁢f⁢ei⁢2⁢π⁢f⁢(−t)¯d f =s⁢−t ℱ S f f S f 2 f t f S f 2 f t s t Therefore, two Fourier transforms applied to s⁢t s t yields s⁢−t s t . We need two more to get us back where we started.

Content actions

Download 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 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