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Propiedades de Simetría de las Series de Fourier

Module by: Justin Romberg. E-mail the authorTranslated By: Fara Meza, Erika Jackson

Based on: Symmetry Properties of the Fourier Series by Justin Romberg

Summary: 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.

Theorem 1

Si ft f t es real eso implica que ft=ft* f t f t ( ft* f t es el complejo conjugado de ft f t ), entonces c n = c - n * c n c - n lo cual implica que Re c n =Re c - n c n c - n , Por ejemplo, la parte real de c n c n es par, y Im c n =Im c - n c n c - n , Por ejemplo, tla parte imaginaria de c n c n es impar. Vea figura 1. Lo que tambien implica que | c n |=| c - n | c n c - n , Por ejemplo, que la magnitud es par, y que la c n =( c - n ) c n c - n , Por ejemplo, el ángulo es impar.

Proof

c - n =1T0Tftej ω 0 ntd t =(1T0Tft*e(j ω 0 nt)d t )*  ,   ft=ft*   =(1T0Tfte(j ω 0 nt)d t )*= c n * 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
(1)

Figura 1: Re c n =Re c - n c n c - n , y Im c n =Im c - n c n c - n .
(a)
Figura 1(a) (m10838ae.png)
(b)
Figura 1(b) (m10838ce.png)
Figura 2: | c n |=| c - n | c n c - n , y c n =( c - n ) c n c - n .
(a)
Figura 2(a) (m10838be.png)
(b)
Figura 2(b) (m10838de.png)

Señales Reales y Pares

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

Theorem 2

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

Proof

c n =1TT2T2fte(j ω 0 nt)d t =1TT20fte(j ω 0 nt)d t +1T0T2fte(j ω 0 nt)d t =1T0T2ftej ω 0 ntd t +1T0T2fte(j ω 0 nt)d t =2T0T2ftcos ω 0 ntd t 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
(2)
ft f t y cos ω 0 nt ω 0 n t son reales lo cual implica que c n c n es real. También cos ω 0 nt=cos( ω 0 nt) ω 0 n t ω 0 n t entonces c n = c - n c n c - n . Es tán fácil demostrar que ft=2 n =0 c n cos ω 0 nt f t 2 n 0 c n ω 0 n t ya que ft f t , c n c n , y cos ω 0 nt 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.

Theorem 3

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

Proof

Hágalo usted en casa.

Si ft f t es impar, podemos expenderlos en términos de sin ω 0 nt ω 0 n t : ft= n =12 c n sin ω 0 nt f t n 1 2 c n ω 0 n t

Resumen

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

ft= f e t+ f o t f t f e t f o t
(3)
lo cual implica, que para cualquier ft f t , podemos encontrar a n a n y b n b n que da
ft= n =0 a n cos ω 0 nt+ n =1 b n sin ω 0 nt f t n 0 a n ω 0 n t n 1 b n ω 0 n t
(4)

Ejemplo 1: La Función Triangular

Figura 3: T=1 T 1 y ω 0 =2π ω 0 2 .
Figura 3 (triwave.png)

ft f t es real e impar. c n ={4Ajπ2n2  if  n=-11-7-31594Ajπ2n2  if  n=-9-5-137110  if  n=-4-2024 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 c n c - n ?

Figura 4: Series de Fourier para una funcion triangular.
Figura 4 (m10838ee.png)

Nota:

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 1n 1 n . La función triangular es continua, pero no es diferenciable, y sus series de Fourier disminuyen como 1n2 1 n 2 .

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

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