# Connexions

You are here: Home » Content » Signals, Systems, and Society » Parseval's Theorem for the Fourier Series

• #### 4. The Laplace Transform

• 5. References for Signals, Systems, and Society

### 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.

Inside Collection (Textbook):

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

# Parseval's Theorem for the Fourier Series

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

Recall that in Chapter 1, we defined the power of a periodic signal as

p x = 1 T t 0 t 0 + T x 2 ( t ) d t p x = 1 T t 0 t 0 + T x 2 ( t ) d t
(1)

where TT is the period. Using the complex form of the Fourier series, we can write

x ( t ) 2 = n = - c n e j n Ω 0 t m = - c m e j m Ω 0 t * x ( t ) 2 = n = - c n e j n Ω 0 t m = - c m e j m Ω 0 t *
(2)

where we have used the fact that x(t)2=x(t)x(t)*x(t)2=x(t)x(t)*, i.e. since x(t)x(t) is real x(t)=x(t)*x(t)=x(t)*. Applying (Reference) and (Reference) gives

x ( t ) 2 = n = - c n e j n Ω 0 t m = - c m * e - j m Ω 0 t = n = - m = - c n c m * e j ( n - m ) Ω 0 t = n = - c n 2 + n m c n c m * e j ( n - m ) Ω 0 t x ( t ) 2 = n = - c n e j n Ω 0 t m = - c m * e - j m Ω 0 t = n = - m = - c n c m * e j ( n - m ) Ω 0 t = n = - c n 2 + n m c n c m * e j ( n - m ) Ω 0 t
(3)

Substituting this quantity into Equation 1 gives

p x = 1 T t 0 t 0 + T n = - c n 2 + n m c n c m * e j ( n - m ) Ω 0 t d t = n = - c n 2 + 1 T t 0 t 0 + T n m c n c m * e j ( n - m ) Ω 0 t d t p x = 1 T t 0 t 0 + T n = - c n 2 + n m c n c m * e j ( n - m ) Ω 0 t d t = n = - c n 2 + 1 T t 0 t 0 + T n m c n c m * e j ( n - m ) Ω 0 t d t
(4)

It is straight-forward to show that

1 T t 0 t 0 + T n m c n c m * e j ( n - m ) Ω 0 t d t = 0 1 T t 0 t 0 + T n m c n c m * e j ( n - m ) Ω 0 t d t = 0
(5)

This leads to Parseval's Theorem for the Fourier series:

p x = n = - c n 2 p x = n = - c n 2
(6)

which states that the power of a periodic signal is the sum of the magnitude of the complex Fourier series coefficients.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### 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?

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?

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.