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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.

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.

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=ftgt 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 . ftgt f t g t es una convolución circular de dos señales periódica y es equivalente a una convolución en el intervalo, ftgt=0T0Tfτgtτdτdt 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 =1T0Tvt- ω 0 ntdt=1T20T0Tfτgtτdτ- ω 0 ntdt=1T0Tfτ1T0Tgtτ- ω 0 ntdtdτ=ν,ν=tτ:1T0Tfτ1T-τTτgν- ω 0 ν+τdνdτ=1T0Tfτ1T-τTτgν- ω 0 nνdν- ω 0 nτdτ=1T0Tfτdn- ω 0 nτdτ= d n 1T0Tfτ- 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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