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Análisis de Fourier

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

Based on: Fourier Analysis by Richard Baraniuk

Summary: Lista de las cuatro transformadas de Fourier y cuando se usan.

El análisis de Fourier es elemental para entender el comportamiento de las señales de sistemas. Este es el resultado de que los senosoidales son eigenfunciones de sistemas lineales variantes en el tiempo (LTI). Si pasamos cualquier senosoidal a través de un sistema LTI, obtenemos la versión escalada de cualquier sistema senosoidal como salida. Entonces, ya que el análisis de Fourier nos permite redefinir las señales en terminos de senosoidales, todo lo que tenemos que hacer es determinar el efecto que cualquier sistema tiene en todos los senosoidales posibles (su función de transferencia) así tendremos un entendimiento completo del sistema. Así mismo, ya que podemos definir el paso de los senosoidales en el sistema como la multiplicación de ese senosoidal por la función de transferencia en la misma frecuencia, puedes convertir el paso de la señal a través de cualquier sistema de ser una convolución (en tiempo) a una multiplicación (en frecuencia) estas ideas son lo que dan el poder al análisis de Fourier.

Ahora, después de haberle vendido el valor que tiene este método de análisis, nosotros debemos analizar exactamente lo que significa el análisis Fourier. Las cuatro transformadas de Fourier que forman parte de este análisis son: Series Fourier, Transformada de Fourier continua en el tiempo, Transformada de Fourier en Tiempo Discreto, y La Transformada de Fourier Discreta. Para este modulo, nosotros veremos la trasformada de Laplace y la transformada Z. Como extensiones de CTFT y DTFT respectivamente. Todas estas transformadas actúan esencialmente de la misma manera, al convertir una señal en tiempo en su señal equivalente en frecuencia (senosoidales). Sin embargo, dependiendo en la naturaleza de una señal especifica (por ejemplo, si es de tamaño finito o infinito, o si son discretas o continuas en el tiempo) hay una transformada apropiada para convertir las señales en su dominio de frecuencia. La siguiente tabla muestra las cuatro transformadas de Fourier y el uso de cada una. También incluye la convolucion relevante para el espacio especificado.

Tabla 1: Tabla de Representaciones para Fourier
Transformada Dominio del Tiempo Dominio de la Frecuencia Convolución
Serie de Fourier Continua en el Tiempo L 2 ⁢ 0 T L 2 0 T l 2 ⁢Z l 2 Tiempo Continuo Circular
Transformada de Fourier en Tiempo Continuo L 2 ⁢R L 2 L 2 ⁢R L 2 Tiempo Continuo Lineal
Transformada de Fourier Discreta en el Tiempo l 2 ⁢Z l 2 L 2 ⁢ 0 2⁢π L 2 0 2 Tiempo Discreto Lineal
Transformada de Fourier Discreta l 2 ⁢ 0 N−1 l 2 0 N 1 l 2 ⁢ 0 N−1 l 2 0 N 1 Tiempo Discreto Circular

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