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Desplazamientos Circulares

Module by: Justin Romberg Translated By Fara Meza, Erika JacksonBased on: Circular Shifts by Justin Romberg

Summary: Este modulo ve el desplazamiento circular y como se puede usar para representar el desplazamiento de secuencias periodicas.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Las muchas propiedades de la DFTS se convierten sencillas (muy similares a las de las Series de Fourier) cuando entendemos el concepto de: desplazamientos circulares.

Desplazamientos Circulares

Podemos describir las secuencias periódicas teniendo puntos discretos en un círculo como su dominio.

Figura 1
Figura 1 (fig01.png)

Desplazar mm, fn+m f n m , corresponde a rotar el cilindro mm puntos ACW (en contra del reloj). Para m=-2 m -2 , obtenemos un desplazamiento igual al que se ve en la siguiente ilustración:

Figura 2: para m=-2 m -2
Figura 2 (fig2.png)
Figura 3
Figura 3 (fig03.png)

Para ciclar los desplazamientos seguiremos los siguientes pasos:

1) Escriba fn f n en el cilindro, ACW

Figura 4: N=8 N 8
Figura 4 (fig4.png)

2) Para ciclar por mm, gire el cilindro m lugares ACW fn→f (( n + m )) N → f n f (( n + m )) N

Figura 5: m=-3 m -3
Figura 5 (fig5.png)

Ejemplo 1

Si fn=01234567 f n 0 1 2 3 4 5 6 7 , entonces f (( n - 3 )) N =34567012 f (( n - 3 )) N 3 4 5 6 7 0 1 2

Es llamado desplazamiento circular, ya que nos estamos moviendo alrededor del círculo. El desplazamiento común es conocido como “desplazamiento linear” (un movimiento en una línea).

Notas para el desplazamiento circular

f (( n + N )) N =fn f (( n + N )) N f n Girar por NN lugares es lo mismo que girar por una vuelta completa, o no moverse del mismo lugar.

f (( n + N )) N =f (( n - ( N - m ) )) N f (( n + N )) N f (( n - ( N - m ) )) N Desplazar ACW mm es equivalente a desplazar CW N−m N m

Figura 6
Figura 6 (fig6.png)

f (( - n )) N f (( - n )) N La expresión anterior, escribe los valores de fn f n para el lado del reloj.

Figura 7
(a) fn f n (b) f (( - n )) N f (( - n )) N
Figura 7(a) (fig7a.png)Figura 7(b) (fig7b.png)

Desplazamientos Circulares y el DFT

Theorem 1: Desplazamiento Circular y el DFT

Si fn ↔ DFT Fk f n ↔ DFT F k entonces f (( n - m )) N ↔ DFT ⅇ-ⅈ2πNkmFk f (( n - m )) N ↔ DFT 2 N k m F k (Por ejemplo. Desplazamiento circular en el dominio del tiempo= desplazamiento del ángulo en el DFT)

Proof

fn=1N∑k=0N−1Fkⅇⅈ2πNkn f n 1 N k 0 N 1 F k 2 N k n (1)
así que el desplazar el ángulo en el DFT
fn=1N∑k=0N−1Fkⅇ-ⅈ2πNknⅇⅈ2πNkn=1N∑k=0N−1Fkⅇⅈ2πNkn−m=f (( n - m )) N f n 1 N k 0 N 1 F k 2 N k n 2 N k n 1 N k 0 N 1 F k 2 N k n m f (( n - m )) N (2)

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