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3.2 Propiedades de la convolución

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Ejemplos y definiciones de varias propiedades asociadas con la convolución son descritas.

En este modulo veremos varias de las propiedades de convolución que mas prevalecen. Nótese que estas propiedades se aplican a ambas convoluciones de tiempo continuo y de tiempo discreto . (Véase los dos módulos anteriores si necesita un repaso de convolución). También para algunas demostraciones de las propiedades, usaremos las integrales de tiempo-continuo, pero podemos probarlas de la misma manera usando las sumatorias de tiempo-discreto.

Asociatividad

Ley asociativa

f 1 t f 2 t f 3 t f 1 t f 2 t f 3 t

Implicación gráfica de la propiedad de asociatividad de la convolución.

Conmutatividad

: ley conmutativa

y t f t h t h t f t

Para probar la , lo único que tenemos que hacer es un pequeño cambio de variable en nuestra integral de convolución (o suma),

y t τ ∞ ∞ f τ h t τ
Dejando τ t τ , podemos mostrar fácilmente que la convolución es conmutativa :
y t τ ∞ ∞ f t τ h τ τ ∞ ∞ h τ f t τ
f t h t h t f t

La figura muestra que ambas funciones pueden ser vistas como entradas del sistema mientras lo otro es la respuesta al impulso.

Distribución

Ley distributiva

f 1 t f 2 t f 3 t f 1 t f 2 t f 1 t f 3 t

La demostración de este teorema puede ser tomada directamente de la definición de convolución y usando la linealidad de la integral.

Desplazamiento en el tiempo

Propiedad de desplazamiento

Para c t f t h t , entonces

c t T f t T h t
y
c t T f t h t T

Demostración Gráfica de la propiedad de desplazamiento.

Convolución con un impulso

Convolución con impulso unitario

f t δ t f t

Para este demostración, dejaremos que δ t sea el impulso unitario localizado en el origen. Usando la definición de convolución empezamos con la integral de convolución

f t δ t τ ∞ ∞ δ τ f t τ
De la definición del impulso unitario, conocemos que δ τ 0 siempre que τ 0 . Usamos este hecho para reducir la ecuación anterior y obtener lo siguiente:
f t δ t τ ∞ ∞ δ τ f t f t τ ∞ ∞ δ τ
La integral de δ τ solo tendrá un valor cuando τ 0 (de la definición del impulso unitario), por lo tanto esa integral será igual a uno. Donde podemos simplificar la ecuación de nuestro teorema:
f t δ t f t

Las figuras y ecuaciones anteriores, revelan la función identidad del impulso unitario.

Ancho

En tiempo continuo, si la Duración f 1 T 1 y la Duración f 2 T 2 , entonces

Duración f 1 f 2 T 1 T 2

En tiempo continuo, la duración de la convolución resulta igual a la suma de las longitudes de cada una de las dos señales convolucionadas.

En tiempo discreto si la Duración f 1 N 1 y la Duración f 2 N 2 , entonces

Duración f 1 f 2 N 1 N 2 1

Causalidad

Si f y h son ambas causales, entonces f h también es causal.

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Source:  OpenStax, Señales y sistemas. OpenStax CNX. Sep 28, 2006 Download for free at http://cnx.org/content/col10373/1.2
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