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Propiedades de la Transformada de Fourier

Module by: Don Johnson. E-mail the authorTranslated By: Fara Meza, Erika Jackson

Based on: Fourier Transform Properties by Don Johnson

Summary: Una tabla de las transformadas mas comunes, para su referencia.

Tabla 1: Pequeña Tabla de los Pares de la Transformada de Fourier
s(t) S(f)
e−(a⁢t)⁢u⁢t a t u t 1i⁢2⁢π⁢f+a 1 2 f a
e(−a)⁢|t| a t 2⁢a4⁢π2⁢f2+a2 2 a 4 2 f 2 a 2
p⁢t={1  if  |t|<Δ20  if  |t|>Δ2 p t 1 t Δ 2 0 t Δ 2 sinπ⁢f⁢Δπ⁢f f Δ f
sin2⁢π⁢W⁢tπ⁢t 2 W t t S⁢f={1  if  |f|<W0  if  |f|>W S f 1 f W 0 f W
Tabla 2: Propiedades de la Transformada de Fourier
Dominio del Tiempo Dominio de la Frecuencia
Linealidad a 1 ⁢ s 1 ⁢t+ a 2 ⁢ s 2 ⁢t a 1 s 1 t a 2 s 2 t a 1 ⁢ S 1 ⁢f+ a 2 ⁢ S 2 ⁢f a 1 S 1 f a 2 S 2 f
Simetria Conjugada s⁢t∈R s t S⁢f=S⁢−f¯ S f S f
Simetria Par s⁢t=s⁢−t s t s t S⁢f=S⁢−f S f S f
Simetria Impar s⁢t=−s⁢−t s t s t S⁢f=−S⁢−f S f S f
Cambio de Escala s⁢a⁢t s a t 1|a|⁢S⁢fa 1 a S f a
Retraso en el Tiempo s⁢t−τ s t τ e−(i⁢2⁢π⁢f⁢τ)⁢S⁢f 2 f τ S f
Modulación Compleja ei⁢2⁢π⁢ f 0 ⁢t⁢s⁢t 2 f 0 t s t S⁢f− f 0 S f f 0
Amplitud Modulada por Coseno s⁢t⁢cos2⁢π⁢ f 0 ⁢t s t 2 f 0 t S⁢f− f 0 +S⁢f+ f 0 2 S f f 0 S f f 0 2
Amplitud Modulada por Seno s⁢t⁢sin2⁢π⁢ f 0 ⁢t s t 2 f 0 t S⁢f− f 0 −S⁢f+ f 0 2⁢i S f f 0 S f f 0 2
Derivación dd t s⁢t t s t i⁢2⁢π⁢f⁢S⁢f 2 f S f
Integración ∫−∞ts⁢αd α α t s α 1i⁢2⁢π⁢f⁢S⁢f 1 2 f S f if S⁢0=0 S 0 0
Multiplicación por tt t⁢s⁢t t s t 1−(i⁢2⁢π)⁢dS⁢fd f 1 2 f S f
Área ∫−∞∞s⁢td t t s t S⁢0 S 0
Valor en el Origen s⁢0 s 0 ∫−∞∞S⁢fd f f S f
Teorema de Parseval ∫−∞∞|s⁢t|2d t t s t 2 ∫−∞∞|S⁢f|2d f f S f 2

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