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Resumen de las Series de Fourier

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

Based on: Fourier Series Wrap-Up by Michael Haag, Justin Romberg

Summary: (Blank Abstract)

Abajo veremos algunos de los conceptos más importantes de las series de Fourier y nuestro entendimiento usando eigenfunciones y eigenvalores. Ojala este familiarizado con este material para que este documento sirva como un repaso, pero si no, use todos los links de información dados en los temas.

  1. Podemos representar una función periódica o una función en un intervalo” f⁢t f t como la combinación de exponenciales complejos:
    f⁢t=∑ n =−∞∞ c n ⁢ej⁢ ω 0 ⁢n⁢t f t n c n ω 0 n t
    (1)
    c n =1T⁢∫0Tf⁢t⁢e−(j⁢ ω 0 ⁢n⁢t)d t c n 1 T t T 0 f t ω 0 n t
    (2)
    Donde los coeficientes de Fourier, c n c n , igualan cuanto de la frecuencia ω 0 ⁢n ω 0 n existen en la señal.
  2. Ya que ej⁢ ω 0 ⁢n⁢t ω 0 n t son eigenfunciones de sistema LTI podemos interpretar la acción de un sistema en una señale en termino de sus eigenvalores:
    H⁢j⁢ ω 0 ⁢n=∫−∞∞h⁢t⁢e−(j⁢ ω 0 ⁢n⁢t)d t H ω 0 n t h t ω 0 n t
    (3)
    • |H⁢j⁢ ω 0 ⁢n| H ω 0 n es grande ⇒ el sistema acentúa la frecuencia ω 0 ⁢n ω 0 n
    • |H⁢j⁢ ω 0 ⁢n| H ω 0 n es pequeño⇒ el sistema atenúa el ω 0 ⁢n ω 0 n
  3. En adición el c n c n de una función periódica f⁢t f t nos puede decir sobre:
    • simetrías en f⁢t f t
    • suavidad en f⁢t f t , where donde la suavidad se puede interpretar como el radio de decadencia | c n | c n .
  4. Podemos aproximar una función a de-sintetizar usando algunos valores en el coeficiente de fourier ( truncando la S.F.)
    f N ′t=∑ n n≤|N| c n ⁢ej⁢ ω 0 ⁢n⁢t f N t n n N c n ω 0 n t
    (4)
    Esta aproximación funciona bien donde f⁢t f t es continuo pero no función también cuando f⁢t f t is discontinuous. es descontinuó esto es explicado por el fenómeno de Gibb.

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