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Extensión Periódica de las DTFS

Module by: Roy Ha. E-mail the authorTranslated By: Fara Meza, Erika Jackson

Based on: Periodic Extension to DTFS by Roy Ha

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

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Introducción

Ya que contamos con entendimiento de lo que son las series discretas de Fourier (DTFS), podemos considerar la extensión periódica de ck 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.

Figura 1
(a) vectores
Figura 1(a) (fig1a.png)
(b) secuencias periodicas
Figura 1(b) (fig1b.png)

Exercise 1

¿Porqué una extensión periódica de los coeficientes de la DFTS ck c k tiene sentido?

Solution

Aliasing: b k =ⅇⅈ2πNkn b k 2 N k n

b k + N =ⅇⅈ2πNk+Nn=ⅇⅈ2πNknⅇⅈ2πn=ⅇⅈ2πNn= b k b k + N 2 N k N n 2 N k n 2 n 2 N n b k (1)
→ Coeficientes DTFS también son periódicos con periodo NN.

Ejemplos

Ejemplo 1: Función Cuadrada Discreta

Figura 2
Figura 2 (fig2.png)

Calcule la DTFS ck c k usando:

ck=1N∑n=0N−1fnⅇ-ⅈ2πNkn c k 1 N n 0 N 1 f n 2 N k n (2)
Como en las series de Fourier continuas, podemos tomar la sumatoria en cualquier intervalo, para tener
c k =1N∑n=- N 1 N 1 ⅇ-ⅈ2πNkn c k 1 N n N 1 N 1 2 N k n (3)
Deje que m=n+ N 1 m n N 1 ,(así tenemos una serie geométrica que empieza en 0)
c k =1N∑m=02 N 1 ⅇ-ⅈ2πNm− N 1 k=1Nⅇⅈ2πNk∑m=02 N 1 ⅇ-ⅈ2πNmk 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 (4)
Ahora, usando la “formula parcial de sumatoria”
∑n=0Man=1−aM+11−a n 0 M a n 1 a M 1 1 a (5)
c k =1Nⅇⅈ2πN N 1 k∑m=02 N 1 ⅇ-ⅈ2πNkm=1Nⅇⅈ2πN N 1 k1−ⅇ-ⅈ2πN2 N 1 +11−ⅇ-ⅈk2πN 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 (6)
Manipulándola para que se vea como el sinc (distribuya)
c k =1Nⅇ-ⅈk2π2Nⅇⅈk2πN N 1 +12−ⅇ-ⅈk2πN N 1 +12ⅇ-ⅈk2π2Nⅇⅈk2πN12−ⅇ-ⅈk2πN12=1Nsin2πk N 1 +12NsinπkN=digital sinc 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 (7)

note:

¡es periódica! figura 3, figura 4, y figura 5 muestran como nuestra función y coeficientes para distintos valores de N 1 N 1 .

Figura 3: N 1 =1 N 1 1
(a) Grafíca de fn f n . (b) Grafíca de ck c k .
Figura 3(a) (dtfs1.png)Figura 3(b) (dtfs1a.png)
Figura 4: N 1 =3 N 1 3
(a) Grafíca de fn f n . (b) Grafíca de ck c k .
Figura 4(a) (dtfs2.png)Figura 4(b) (dtfs2a.png)
Figura 5: N 1 =7 N 1 7
(a) Grafíca de fn f n . (b) Grafíca de ck c k .
Figura 5(a) (dtfs3.png)Figura 5(b) (dtfs3a.png)

Ejemplo 2: Sonido de un Pájaro

Figura 6
Figura 6 (nofig.png)

Ejemplo 3: Análisis Espectral de la DFTS

Figura 7
Figura 7 (nofig.png)

Ejemplo 4: Recuperando la Señal

Figura 8
Figura 8 (nofig.png)

Ejemplo 5: Compresión (1-D)

Figura 9
Figura 9 (nofig.png)

Ejemplo 6: Compresión de Imágenes

Figura 10: We've cut down on storage space by > 90% (while incurring some loss)
(a) Imagen de 256 por 256 (65,636 pixeles)
Figura 10(a) (nofig.png)
(b) Imagen reconstruida usando 6000 coefícientes
Figura 10(b) (nofig.png)

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