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Muestreo

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

Based on: Sampling by Justin Romberg

Summary: Este modulo trata con la traslación de problemas de tiempo continuo a problemas de tiempo discreto.

Introducción

Una computadora puede procesar señales de tiempo discreto usando un lagoritmo extremandamente flexible y poderoso. Mas sin embargo la mayoria de las señales de interes son de tiempo continuo, que es como casi siempre aparecen al natural.

Este modulo introduce la idea de trasladar los problemas de tiempo continuo en unos de tiempo discreto, y podra leer más de los detalles de la importancia de el muestreo.

Preguntas clave

  • ¿Cómo pasamos de una señal de tiempo continuo a una señal de tiempo discreto (muestreo, A/D)?
  • ¿Cuándo podemos reconstruir una señal CT exacta de sus muestras (reconstrucción, D/A)?
  • ¿Manipular la señal DT es lo que reconstruir la señal?

Muestreo

Muestreo (y reconstrucción) son los mejores entendimiento en dominio de frecuencia. Empezaremos viendo algunos ejemplos:

Exercise 1

¿Qué señal CT ft f t tiene la CTFT mostrada anterirormente? ft=12πFjwejwtd w f t 1 2 w F w w t

Figura 1: La (Transformada de Fourier de Tiempo Continuo)CTFT de ft f t .
Figura 1 (samp1.png)

Pista: Fjw= F 1 jw* F 2 jw F w F 1 w F 2 w donde las dos partes de Fjw F w son:

Figura 2
(a) (b)
Figura 2(a) (samp2.png)Figura 2(b) (samp3.png)

Solution

ft=12πFjwejwtd w f t 1 2 w F w w t

Exercise 2

¿Qué señal DT f s n f s n tiene la DTFT mostrada anteriormente? f s n=12πππ f s ejwejwnd w f s n 1 2 w f s w w n

Figura 3: DTFT que es périodica (con period=2π period 2 ) versión de Fjw F w en la figura 1.
Figura 3 (samp4.png)

Solution

Ya que Fjw=0 F w 0 afuera de -2 2 -2 2 ft=12π-22Fjwejwtd w f t 1 2 w -2 2 F w w t También , ya que solo utilizamos un intervalo para reconstruir f s n f s n de su DTFT, tenemos f s n=12π-22 f s ejwejwnd w f s n 1 2 w -2 2 f s w w n Ya que Fjw= F s ejw F w F s w en -2 2 -2 2 f s n=ft| t =n f s n t n f t es decir f s n f s n es una versión muestreada de ft f t .

Figura 4: ft f t es la señal de tiempo-continuo anterior y f s n f s n es la versión muestreada de tiempo-discreto de ft f t
Figura 4 (samp_big.png)

Generalización

Por supuesto, que los resultados de los ejemplos anteriores pueden ser generalizados a cualquier ft f t con Fjw=0 F w 0 , |w|>π w , donde ft f t es limitado en banda a π π .

Figura 5: Fjw F w es la CTFT de ft f t .
(a) (b)
Figura 5(a) (samp_g1.png)Figura 5(b) (samp_g2.png)
Figura 6: F s ejw F s w es la DTFT de f s n f s n .
(a) (b)
Figura 6(a) (samp_g3.png)Figura 6(b) (samp_g4.png)

F s ejw F s w es períodico (con período 2π 2 ) versión de Fjw F w . F s ejw F s w es la DTFT de muestreo de señal en los enteros. Fjw F w es la CTFT de señal.

conclusion:

Si ft f t es limitado en banda para π π entonces la DTFT de la versión muestreada f s n=fn f s n f n es solo periódica (con período 2π 2 ) versión de Fjw F w .

Cambiando una Señal Discreta en una Señal Continua

Ahora veamos como cambiar una señal DT en una señal continua en el tiempo. Sea f s n f s n una señal DT con DTFT F s ejw F s w

Figura 7: F s ejw F s w es la DTFT de f s n f s n .
(a) (b)
Figura 7(a) (samp_e1.png)Figura 7(b) (samp_e2.png)

Ahora, sea f imp t= n = f s nδtn f imp t n f s n δ t n La señal CT, f imp t f imp t , es no-cero solo en los enteros donde hay implulsos de altura f s n f s n .

Figura 8
Figura 8 (samp_e3.png)

Exercise 3

¿Cúal es la CTFT de f imp t f imp t ?

Solution

f imp t= n = f s nδtn f imp t n f s n δ t n

F imp jw= f imp te(jwt)d t = n = f s nδtne(jwt)d t = n = f s nδtne(jwt)d t = n = f s ne(jwn)= F s ejw F imp w t f imp t w t t n f s n δ t n w t n f s n t δ t n w t n f s n w n F s w
(1)

Así que la CTFT de f imp t f imp t es igual a la DTFT de f s n f s n

nota:
Usamos la propiedad de desplazamiento para mostrar δtne(jwt)d t =e(jwn) t δ t n w t w n

Ahora, dadas las muestras f s n f s n de un limitado en banda para la señal π π , nuestro siguiente paso es ver como podemos reconstruir ft f t .

Figura 9: Diagrama de bloque mostrando cada paso básico usado para reconstruir ft f t . ¿Podemos hacer nuestro resultado igual a ft f t exactamente?
Figura 9 (samp_blka.png)

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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

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