Skip to content Skip to navigation Skip to collection information

OpenStax CNX

You are here: Home » Content » Señales y Sistemas » Convolución Circular y el DFT

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.
Download
x

Download collection as:

  • PDF
  • EPUB (what's this?)

    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 "(what's this?)" link.

  • More downloads ...

Download module as:

  • PDF
  • EPUB (what's this?)

    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 "(what's this?)" link.

  • More downloads ...
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:

 

Convolución Circular y el DFT

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

Based on: Circular Convolution and the DFT by Justin Romberg

Summary: Este modulo describe el elgoritmo de convolucion cicular y un algoritmo alterno

Introducción

Usted debería familiarizarse con la convolución discreta, que nos explica como dos señales discretas xn x n , la entrada del sistema, y hn h n , la respuesta del sistema, se puede definir el resultado del sistema como

yn=xn*hn= k =xkhnk y n x n h n k x k h n k
(1)
Cuando dos DFT son dadas (secuencias de tamaño finito usualmente del tamaño NN), nosotros no podemos multiplicar esas dos señales así como así, como lo sugiere la formula de arriba usualmente conocida como convolución linear. Ya que las DFT son periódicas, tienen valores no cero para nN n N así la multiplicación de estas dos señales será no cero para nN n N . Necesitamos definir otro tipo de convolucion que dará como resultado nuestra señal convuelta teniendo el valor de cero fuera del rango n=01N1 n 0 1 N 1 . Esto nos ayuda a desarrollar la idea de convolución circular, también conocida como convolución cíclica o periódica.

Formula de la Convolución Circular

¿Qué pasa cuando multiplicamos dos DFT una con la otra, donde Yk Y k es la DFT de yn y n ?

Yk=FkHk Y k F k H k
(2)
cuando 0kN1 0 k N 1

Usando la formula sintetizada de DFT para yn y n

yn=1N k =0N1FkHkej2πNkn y n 1 N k 0 N 1 F k H k j 2 N k n
(3)

Y aplicando análisis a la formula Fk= m =0N1fme(j)2πNkn F k m 0 N 1 f m j 2 N k n

yn=1N k =0N1 m =0N1fme(j)2πNknHkej2πNkn= m =0N1fm(1N k =0N1Hkej2πNk(nm)) y n 1 N k 0 N 1 m 0 N 1 f m j 2 N k n H k j 2 N k n m 0 N 1 f m 1 N k 0 N 1 H k j 2 N k n m
(4)
donde podemos reducir la segunda sumatoria de la ecuación de arriba en h ( ( n m ) ) N =1N k =0N1Hkej2πNk(nm) h ( ( n m ) ) N 1 N k 0 N 1 H k j 2 N k n m yn= m =0N1fmh ( ( n m ) ) N y n m 0 N 1 f m h ( ( n m ) ) N Igual a la convolución circular! cuando tenemos 0nN1 0 n N 1 arriba , para obtener dos:
ynfnhn y n f n h n
(5)

note:

Que la notación representa la convolucion circular "mod N".

Pasos para la Convolución Circular

Los pasos a seguir para la convolucion cíclica son los mismos que se usan en la convolución linear, excepto que todos los cálculos para todos los índices están hecho"mod N" = "en la rueda"

Pasos para la Convolución Cíclica

  • Paso 1: "Grafique" fm f m y h (((m()() N h ( ( m ) ) N
Figura 1: Step 1
(a) (b)
Figura 1(a) (cconv_s1.png)Figura 1(b) (cconv_s2.png)
  • Paso 2: "Rote" h (((m()() N h ( ( m ) ) N n n en la dirección ACW ( dirección opuesta al reloj) para obtener h ((n((m()() N h ( ( n m ) ) N (por ejemplo rote la secuencia, hn h n , en dirección del reloj por nn pasos).
Figura 2: Step 2
Figura 2 (cconv_s3.png)
  • Paso 3: Multiplique punto por punto la rueda fm f m y la rueda h ((n((m()() N h ( ( n m ) ) N wheel. sum=yn sum y n
  • Paso 4: Repite para 0nN1 0 n N 1

Ejemplo 1: Convolve (n = 4)

Figura 3: Dos señales discretas que seran convolucionadas.
(a) (b)
Figura 3(a) (cconv_p1.png)Figura 3(b) (cconv_p2.png)

  • h (((m()() N h ( ( m ) ) N

Figura 4
Figura 4 (cconv_p3.png)

Multiplique fm f m y sume sume para dar: y0=3 y 0 3

  • h ((1((m()() N h ( ( 1 m ) ) N

Figura 5
Figura 5 (cconv_p4.png)

Multiplique fm f m y sume sume para dar: y1=5 y 1 5

  • h ((2((m()() N h ( ( 2 m ) ) N

Figura 6
Figura 6 (cconv_p5.png)

Multiplique fm f m y sume sume para dar: y2=3 y 2 3

  • h ((3((m()() N h ( ( 3 m ) ) N

Figura 7
Figura 7 (cconv_p6.png)

Multiplique fm f m y sume sume para dar: y3=1 y 3 1

Ejemplo 2

La Siguiente Demostración le permite este algoritmo. Vea aquí para instrucciones de como se usa este demo.

LabVIEW Example: (run) (source)

Algoritmo Alterno

Algoritmo de Convolución Circular Alterno

  • Paso 1: Calcule el DFT de fn f n que da Fk F k y calcule el DFT de hn h n que da Hk H k .
  • Paso 2: Multiplique punto por punto Yk=FkHk Y k F k H k
  • Paso 3: Invierta el DFT Yk Y k que da yn y n

Parece una manera repetitiva de hacer las cosas, pero existen maneras rápidas de calcular una secuencia DFT.

Para convolucionar circularmente dos secuencias de 2 2 N N-puntos: yn= m =0N1fmh ( ( n m ) ) N y n m 0 N 1 f m h ( ( n m ) ) N Para cualquier n n : N N múltiplos, N1 N 1 sumas

N N puntos implica N2 N 2 multiplicaciones, N(N1) N N 1 sumas implica una complejidad de ON2 O N 2 .

Collection Navigation

Content actions

Download:

Collection 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 ...

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:

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.