# Connexions

You are here: Home » Content » Plancharel and Parseval's Theorems

### 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.

# Plancharel and Parseval's Theorems

Module by: Justin Romberg. E-mail the author

Summary: This module contains the definition of the Plancharel theorem and Parseval's theorem along with proofs and examples.

Note: You are viewing an old version of this document. The latest version is available here.

## Plancharel Theorem

### Theorem 1: Plancharel Theorem

The inner product of two vectors/signals is the same as the 2 2 inner product of their expansion coefficients.

Let b i b i be an orthonormal basis for a Hilbert Space H H. xH x H , yH y H x= i α i b i x i α i b i y= i β i b i y i β i b i then x,y H = i α i β i ¯ x y H i α i β i

#### Example

Applying the Fourier Series, we can go from ft f t to c n c n and gt g t to d n d n 0Tftgt¯d t = n = c n d n ¯ t 0 T f t g t n c n d n inner product in time-domain = inner product of Fourier coefficients.

#### Proof

x= i α i b i x i α i b i y= j β j b j y j β j b j x,y H = i α i b i , j β j b j = i α i ( b i , j β j b j )= i α i j β j ¯( b i , b j )= i α i β i ¯ x y H i α i b i j β j b j i α i b i j β j b j i α i j β j b i b j i α i β i by using inner product rules

##### note:
b i , b j =0 b i b j 0 when ij i j and b i , b j =1 b i b j 1 when i=j i j

If Hilbert space H has a ONB, then inner products are equivalent to inner products in 2 2 .

All H with ONB are somehow equivalent to 2 2 .

##### point of interest:
square-summable sequences are important

## Parseval's Theorem

### Theorem 2: Parseval's Theorem

Energy of a signal = sum of squares of it's expansion coefficients

Let xH x H , b i b i ONB

x= i α i b i x i α i b i Then xH2= i | α i |2 H x 2 i α i 2

#### Proof

Directly from Plancharel xH2= x,x H = i α i α i ¯= i | α i |2 H x 2 x x H i α i α i i α i 2

#### Example

Fourier Series 1Tei w 0 nt 1 T w 0 n t ft=1T n c n 1Tei w 0 nt f t 1 T n c n 1 T w 0 n t 0T|ft|2d t = n =| c n |2 t 0 T f t 2 n c n 2

## Content actions

### Give feedback:

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?

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