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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 f⁢t f t es real eso implica que f⁢t=f⁢t* f t f t ( f⁢t* f t es el complejo conjugado de f⁢t 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 =1T⁢∫0Tf⁢t⁢ej⁢ ω 0 ⁢n⁢td t =(1T⁢∫0Tf⁢t*⁢e−(j⁢ ω 0 ⁢n⁢t)d t )*  ,   f⁢t=f⁢t*   =(1T⁢∫0Tf⁢t⁢e−(j⁢ ω 0 ⁢n⁢t)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 f⁢t=f⁢t* f t f t y f⁢t=f⁢−t 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 =1T⁢∫−T2T2f⁢t⁢e−(j⁢ ω 0 ⁢n⁢t)d t =1T⁢∫−T20f⁢t⁢e−(j⁢ ω 0 ⁢n⁢t)d t +1T⁢∫0T2f⁢t⁢e−(j⁢ ω 0 ⁢n⁢t)d t =1T⁢∫0T2f⁢−t⁢ej⁢ ω 0 ⁢n⁢td t +1T⁢∫0T2f⁢t⁢e−(j⁢ ω 0 ⁢n⁢t)d t =2T⁢∫0T2f⁢t⁢cos ω 0 ⁢n⁢td 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)
f⁢t f t y cos ω 0 ⁢n⁢t ω 0 n t son reales lo cual implica que c n c n es real. También cos ω 0 ⁢n⁢t=cos−( ω 0 ⁢n⁢t) ω 0 n t ω 0 n t entonces c n = c - n c n c - n . Es tán fácil demostrar que f⁢t=2⁢∑ n =0∞ c n ⁢cos ω 0 ⁢n⁢t f t 2 n 0 c n ω 0 n t ya que f⁢t f t , c n c n , y cos ω 0 ⁢n⁢t 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 f⁢t=−f⁢−t f t f t y f⁢t=f⁢t* 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 f⁢t f t es impar, podemos expenderlos en términos de sin ω 0 ⁢n⁢t ω 0 n t : f⁢t=∑ n =1∞2⁢ c n ⁢sin ω 0 ⁢n⁢t 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

f⁢t= f e ⁢t+ f o ⁢t f t f e t f o t
(3)
lo cual implica, que para cualquier f⁢t f t , podemos encontrar a n a n y b n b n que da
f⁢t=∑ n =0∞ a n ⁢cos ω 0 ⁢n⁢t+∑ n =1∞ b n ⁢sin ω 0 ⁢n⁢t 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 =2⁢π ω 0 2 .
Figura 3 (triwave.png)

f⁢t f t es real e impar. c n ={4⁢Aj⁢π2⁢n2  if  n=…-11-7-3159…−4⁢Aj⁢π2⁢n2  if  n=…-9-5-13711…0  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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