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Este modulo ve las diferentes propiedades de simetria de las series de fourier y de sus coefficientes.

Propiedades de simetría

Señales reales

Señales reales tienen una serie de fourier con un conjugado simétrico.

Si f t es real eso implica que f t f t ( f t es el complejo conjugado de f t ), entonces c n c - n lo cual implica que c n c - n , Por ejemplo , la parte real de c n es par, y c n c - n , Por ejemplo , tla parte imaginaria de c n es impar. Vea . Lo que tambien implica que c n c - n , Por ejemplo, que la magnitud es par, y que la c n c - n , Por ejemplo , elángulo es impar.

c - n 1 T t 0 T f t ω 0 n t t f t f t 1 T t 0 T f t ω 0 n t 1 T t 0 T f t ω 0 n t c n

c n c - n , y c n c - n .
c n c - n , y c n c - n .

Señales reales y pares

Las señales reales y pares tienen series de fourier que son pares y reales.

If f t f t y f t f t , Por ejemplo , las señal es real y par, entonces entonces c n c - n y c n c n .

c n 1 T t T 2 T 2 f t ω 0 n t 1 T t T 2 0 f t ω 0 n t 1 T t 0 T 2 f t ω 0 n t 1 T t 0 T 2 f t ω 0 n t 1 T t 0 T 2 f t ω 0 n t 2 T t 0 T 2 f t ω 0 n t
f t y ω 0 n t son reales lo cual implica que c n es real. También ω 0 n t ω 0 n t entonces c n c - n . Es tán fácil demostrar que f t 2 n 0 c n ω 0 n t ya que f t , c n , y ω 0 n t son reales y pares.

Señales reales e impares

Señales reales e impares tienen series de fourier que son impares y completamente imaginarias.

Si f t f t y f t f t , Por ejemplo , la señal es real y impar, entonces c n c - n y c n c n , Por ejemplo, c n es impar y completamente imaginaria.

Hágalo usted en casa.

Si f t es impar, podemos expenderlos en términos de ω 0 n t : f t n 1 2 c n ω 0 n t

Resumen

Podemos encontrar f e t , una función par, y f o t , una función impar, por que

f t f e t f o t
lo cual implica, que para cualquier f t , podemos encontrar a n y b n que da
f t n 0 a n ω 0 n t n 1 b n ω 0 n t

La función triangular

T 1 y ω 0 2 .

f t es real e impar. c n 4 A 2 n 2 n -11 -7 -3 1 5 9 4 A 2 n 2 n -9 -5 -1 3 7 11 0 n -4 -2 0 2 4 ¿Es c n c - n ?

Series de Fourier para una funcion triangular.
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Usualmente podemos juntar información sobre la suavidad de una señal al examinar los coeficientes de Fourier.
Hecha un vistazo a los ejemplos anteriores. Las funciones del pulso y sawtooth no son continuas y sus series de Fourier disminuyen como 1 n . La función triangular es continua, pero no es diferenciable, y sus series de Fourier disminuyen como 1 n 2 .

Las siguientes 3 propiedades nos darán una mejor idea de esto.

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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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