Skip to content Skip to navigation Skip to collection information

Connexions

You are here: Home » Content » Señales y Sistemas » Teoremas de Plancharel y Parseval

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.
  • Rice Digital Scholarship

    This collection is included in aLens by: Digital Scholarship at Rice University

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

  • Featured Content display tagshide tags

    This collection is included inLens: Connexions Featured Content
    By: Connexions

    Comments:

    "Señales y Sistemas is a Spanish translation of Dr. Rich Baraniuk's collection Signals and Systems (col10064). The translation was coordinated by an an assistant electrical engineering professor […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

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.

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.
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:

 

Teoremas de Plancharel y Parseval

Module by: Justin Romberg. E-mail the authorTranslated By: Fara Meza, Erika Jackson

Based on: Plancharel and Parseval's Theorems by Justin Romberg

Summary: Este modulo contiene la definición del teorema de Plancharel y del teorema de Parseval con sus demostraciones y ejemplos.

Teorema de Plancharel

Theorem 1: Teorema de Plancharel

El producto interno de dos vectores/señales es el mismo que en ℓ2 ℓ 2 el producto interno de su expansión de coeficientes.

Sea b i b i una base ortonormal para un Espacio de Hilbert H H. x∈H x H , y∈H 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 entonces 〈x,y〉 H =∑ i α i ⁢ β i * x y H i α i β i

Ejemplo

Aplicando las Series de Fourier, podemos ir de f⁢t f t a c n c n y de g⁢t g t a d n d n ∫0Tf⁢t⁢g⁢t*d t =∑ n =−∞∞ c n ⁢ d n * t 0 T f t g t n c n d n el producto interno en el dominio-tiempo = producto interno de los coefientes de Fourier.

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 usando las reglas del producto interno.

nota:
〈 b i , b j 〉=0 b i b j 0 cuando i≠j i j y 〈 b i , b j 〉=1 b i b j 1 cuando i=j i j

Si el espacio de Hillbert H tiene un ONB, los productos internos son equivalentes a los productos internos en ℓ2 ℓ 2 .

Todo H con ONB son de alguna manera equivalente a ℓ2 ℓ 2 .

punto de interes:
las secuencias de cuadrados sumables son importantes.

Teorema de Parseval

Theorem 2: Teorema de Parseval

La energía de una señal = suma de los cuadrados de su expansión de coeficientes.

Sea x∈H x H , b i b i ONB

x=∑ i α i ⁢ b i x i α i b i Entonces ∥x∥H2=∑ i | α i |2 H x 2 i α i 2

Proof

Directamente de Plancharel ∥x∥H2= 〈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

Ejemplo

Series de Fourier 1T⁢ej⁢ w 0 ⁢n⁢t 1 T w 0 n t f⁢t=1T⁢∑ n c n ⁢1T⁢ej⁢ w 0 ⁢n⁢t f t 1 T n c n 1 T w 0 n t ∫0T|f⁢t|2d t =∑ n =−∞∞| c n |2 t 0 T f t 2 n c n 2

Collection Navigation

Content actions

Download module as:

PDF | 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.