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Transformada de Fourier de Tiempo Continuo (CTFT)

Module by: Richard Baraniuk, Melissa Selik. E-mail the authorsTranslated By: Fara Meza, Erika Jackson

Based on: Continuous-Time Fourier Transform (CTFT) by Richard Baraniuk, Melissa Selik

Summary: Detalles de la la transformada de Fourier de Tiempo-Continuo.

Introducción

Debido al gran número de señales de tiempo-continuo que estan presentes en las series de Fourier nos da una primera ojeada de cuantas maneras podemos representar algunas de estas señales de manera general: como una superposición de un nómero de señales senosoidales. Ahora podemos ver la manera de representar señales noperiodicas de tiempo continuo usando la misma idea de superposición. A continuación presentaremos la Transformada de Fourier de Tiempo-Continuo (CTFT), también conocida solo como Transformada de Fourier (FT). Por que la CTFT ahora trataremos con señales no periodicas, encontraremos una manera de incluir todaslas frecuencias en ecuaciones en general.

Ecuaciones

Transformada de Fourier de Tiempo-Continuo

ℱ⁢Ω=∫−∞∞f⁢t⁢e−(j⁢Ω⁢t)d t ℱ Ω t f t Ω t
(1)

Inversa de la CTFT

f⁢t=12⁢π⁢∫−∞∞ℱ⁢Ω⁢ej⁢Ω⁢td Ω f t 1 2 Ω ℱ Ω Ω t
(2)

precaución:

No se confunda con la notación - es común ver la formula anterior escrita un poco diferente. Una de las diferencias más comunes echa por los profesores es la forma de escribir el exponente. Arriba escribimos la variable de la frecuancia readial Ω Ω en el exponencial, donde Ω=2⁢π⁢f Ω 2 f , pero también vemos que los profesores incluyen la expresión más explicicta, j⁢2⁢π⁢f⁢t 2 f t , en el exponencial. Véase aqui para una descripción de la notación utilizada en los modulos de Procesamiento Digital de Señales DSP.

La ecuacuión anterior para las CFT y su inversa vienen directamente de las series de Fourier y de nuesro entendimiento de sus coeficientes. Para la CTFT simplemente utilizamos la intergración en lugar de la simulación para ser capaces de expresar las señales periódicas. Esto debería tener sentido ya que simlemente estamos extendiendo las ideas de las series de Fourier para las CTFT para incluir las señales no-periódicas,y así todo el espectro de la frecuencia. Véase la Derivación de la Transformada de Fourier para una mirada más profunda del tema.

Espacios Relevantes

El mapeo de la Transformada de Fourier de Tiempo-Continuo de longitud-infinita, en señales de tiempo-continuo L2L2 a longitud-infinita,señales de frecuancia-continua en L2L2. Revisando el Análisis de Fourier para una descripción de todos los espacios usados en el análisis de Fourier.

Figura 1: Mapeando L 2 ⁢R L 2 en el dominio del tiempo a L2⁢R L2 en el dominio de frecuencia.
Figura 1 (CTFTspacee.png)

Problemas de Ejemplo

Exercise 1

Encontrar la Transformada de Fourier(CTFT) de la función

f⁢t={e−(α⁢t)  if  t≥00  otherwise   f t α t t 0 0
(3)

[ Show Solution ]

Exercise 2

Encontrar la inversa de la Transformada de Fourier de la onda cuadrada definda como:

X⁢Ω={1  if  |Ω|≤M0  otherwise   X Ω 1 Ω M 0
(6)

[ Show Solution ]

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