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Propiedad de Convolución Circular de las Series de Fourier

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

Based on: Circular Convolution Property of Fourier Series by Justin Romberg

Summary: Este modulo ve la relacion basica de la convolución circular entre dos conjuntos de coeficientes de Fourier.

Convolución Singular de una Señal

Dada a una señal f⁢t f t con coeficientes de Fourier c n c n y una señal g⁢t g t con coeficientes de Fourier d n d n , podemos definir una señal, v⁢t v t , donde v⁢t=f⁢t⊛g⁢t v t ⊛ f t g t Encontramos que la representación de las series de Fourier para y⁢t y t , a n a n , esta que a n = c n ⁢ d n a n c n d n . f⁢t⊛g⁢t ⊛ f t g t es una convolución circular de dos señales periódica y es equivalente a una convolución en el intervalo, f⁢t⊛g⁢t=∫0T∫0Tf⁢τ⁢g⁢t−τd τ d t ⊛ f t g t t 0 T τ 0 T f τ g t τ .

note:

Convulución circular en el dominio del tiempo es equivalente a la multiplicación de coeficientes de fourier.
La prueba es la siguiente
a n =1T⁢∫0Tv⁢t⁢e−(i⁢ ω 0 ⁢n⁢t)d t =1T2⁢∫0T∫0Tf⁢τ⁢g⁢t−τd τ ⁢e−(i⁢ ω 0 ⁢n⁢t)d t =1T⁢∫0Tf⁢τ⁢(1T⁢∫0Tg⁢t−τ⁢e−(i⁢ ω 0 ⁢n⁢t)d t )d τ =∀ ν ,ν=t−τ:1T⁢∫0Tf⁢τ⁢(1T⁢∫−τT−τg⁢ν⁢e−(i⁢ ω 0 ⁢(ν+τ))d ν )d τ =1T⁢∫0Tf⁢τ⁢(1T⁢∫−τT−τg⁢ν⁢e−(i⁢ ω 0 ⁢n⁢ν)d ν )⁢e−(i⁢ ω 0 ⁢n⁢τ)d τ =1T⁢∫0Tf⁢τ⁢dn⁢e−(i⁢ ω 0 ⁢n⁢τ)d τ = d n ⁢(1T⁢∫0Tf⁢τ⁢e−(i⁢ ω 0 ⁢n⁢τ)d τ )= c n ⁢ d n a n 1 T t 0 T v t ω 0 n t 1 T 2 t 0 T τ 0 T f τ g t τ ω 0 n t 1 T τ 0 T f τ 1 T t 0 T g t τ ω 0 n t ν ν t τ 1 T τ 0 T f τ 1 T ν τ T τ g ν ω 0 ν τ 1 T τ 0 T f τ 1 T ν τ T τ g ν ω 0 n ν ω 0 n τ 1 T τ 0 T f τ d n ω 0 n τ d n 1 T τ 0 T f τ ω 0 n τ c n d n
(1)

Ejemplo 1

Vea el pulso cuadrado con periodo, T 1 =T4 T 1 T 4 :

Figura 1
Figura 1 (sqpulse.png)

Para esta señal c n ={1T  if  n=012⁢sinπ2⁢nπ2⁢n  otherwise   c n 1 T n 0 1 2 2 n 2 n

Exercise 1

¿Que señal tiene los coeficientes de Fourier a n = c n 2=14⁢sin2π2⁢nπ2⁢n2 a n c n 2 1 4 2 n 2 2 n 2 ?

Solution

Figura 2: Un pulso triangular con periodo de T4 T 4 .
Figura 2 (exfig.png)

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