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Propiedad de Convolución Circular de las Series de Fourier

Module by: Justin Romberg Translated By Fara Meza, Erika JacksonBased 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.

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Convolución Singular de una Señal

Dada a una señal ft f t con coeficientes de Fourier c n c n y una señal gt g t con coeficientes de Fourier d n d n , podemos definir una señal, vt v t , donde vt=ft⊛gt v t ⊛ f t g t Encontramos que la representación de las series de Fourier para yt y t , a n a n , esta que a n = c n d n a n c n d n . ft⊛gt ⊛ f t g t es una convolución circular de dos señales periódica y es equivalente a una convolución en el intervalo, ft⊛gt=∫0T∫0Tfτgt−τdτdt ⊛ 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∫0Tvtⅇ-ⅈ ω 0 ntdt=1T2∫0T∫0Tfτgt−τdτⅇ-ⅈ ω 0 ntdt=1T∫0Tfτ1T∫0Tgt−τⅇ-ⅈ ω 0 ntdtdτ=∀ν,ν=t−τ:1T∫0Tfτ1T∫-τT−τgνⅇ-ⅈ ω 0 ν+τdνdτ=1T∫0Tfτ1T∫-τT−τgνⅇ-ⅈ ω 0 nνdνⅇ-ⅈ ω 0 nτdτ=1T∫0Tfτdnⅇ-ⅈ ω 0 nτdτ= d n 1T∫0Tfτⅇ-ⅈ ω 0 nτdτ= 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 =1Tifn=012sinπ2nπ2notherwise 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=14sin2π2nπ2n2 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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