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Este Modulo ve la extensión Ppriódica de los coeficientes de la DTFS y como los coefcientes juegan un papel critico para manipular las señales.

IntroducciÓN

Ya que contamos con entendimiento de lo que son las series discretas de Fourier (DTFS) , podemos considerar la extensión periódica de c k (coeficientes discretos de Fourier). Las figures básicas mostradas a continuación muestran una simple ilustración de como nosotros podríamos representar una secuencia en forma de señales periódicas graficadas sobre un numero infinito de intervalos.

vectores
secuencias periodicas

¿Porquéuna extensión periódica de los coeficientes de la DFTS c k tiene sentido?

Aliasing: b k 2 N k n

b k + N 2 N k N n 2 N k n 2 n 2 N n b k
→Coeficientes DTFS también son periódicos con periodo N .

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Ejemplos

FunciÓN cuadrada discreta

Calcule la DTFS c k usando:

c k 1 N n 0 N 1 f n 2 N k n
Como en las series de Fourier continuas, podemos tomar la sumatoria en cualquier intervalo, para tener
c k 1 N n N 1 N 1 2 N k n
Deje que m n N 1 ,(asítenemos una serie geométrica que empieza en 0)
c k 1 N m 0 2 N 1 2 N m N 1 k 1 N 2 N k m 0 2 N 1 2 N m k
Ahora, usando la“formula parcial de sumatoria”
n 0 M a n 1 a M 1 1 a
c k 1 N 2 N N 1 k m 0 2 N 1 2 N k m 1 N 2 N N 1 k 1 2 N 2 N 1 1 1 k 2 N
Manipulándola para que se vea como el sinc (distribuya)
c k 1 N k 2 2 N k 2 N N 1 1 2 k 2 N N 1 1 2 k 2 2 N k 2 N 1 2 k 2 N 1 2 1 N 2 k N 1 1 2 N k N digital sinc
¡es periódica! , , y muestran como nuestra función y coeficientes para distintos valores de N 1 .

Grafíca de f n .
Grafíca de c k .
N 1 1
Grafíca de f n .
Grafíca de c k .
N 1 3
Grafíca de f n .
Grafíca de c k .
N 1 7
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AnÁLisis espectral de la dfts

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CompresiÓN de imÁGenes

Imagen de 256 por 256 (65,636 pixeles)
Imagen reconstruida usando 6000 coefícientes
We've cut down on storage space by>90% (while incurring some loss)
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Source:  OpenStax, Señales y sistemas. OpenStax CNX. Sep 28, 2006 Download for free at http://cnx.org/content/col10373/1.2
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