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Fourier Series Wrap-Up

Module by: Michael Haag, Justin Romberg. E-mail the authors

Summary: (Blank Abstract)

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Below we will highlight some of the most important concepts about the Fourier Series and our understanding of it through eigenfunctions and eigenvalues. Hopefully you are familiar with all of this material, so this document will simply serve as a refresher, but if not, then refer to the many links below for more information on the various ideas and topics.

  1. We can represent a periodic function (or a function on an interval) f⁢t f t as a combination of complex exponentials:
    f⁢t=∑ n =−∞∞ c n ⁢ei⁢ ω 0 ⁢n⁢t f t n c n ω 0 n t
    (1)
    c n =1T⁢∫0Tf⁢t⁢e−(i⁢ ω 0 ⁢n⁢t)d t c n 1 T t T 0 f t ω 0 n t
    (2)
    Where the fourier coefficients, c n c n , approximately equal how much of frequency ω 0 ⁢n ω 0 n is in the signal.
  2. Since ei⁢ ω 0 ⁢n⁢t ω 0 n t are eigenfunctions of LTI systems, we can interpret the action of a system on a signal in terms of its eigenvalues:
    H⁢i⁢ ω 0 ⁢n=∫−∞∞h⁢t⁢e−(i⁢ ω 0 ⁢n⁢t)d t H ω 0 n t h t ω 0 n t
    (3)
    • |H⁢i⁢ ω 0 ⁢n| H ω 0 n is large ⇒ system accentuates frequency ω 0 ⁢n ω 0 n
    • |H⁢i⁢ ω 0 ⁢n| H ω 0 n is small ⇒ system attenuates frequency ω 0 ⁢n ω 0 n
  3. In addition, the c n c n of a periodic function f⁢t f t can tell us about:
    • symmetries in f⁢t f t
    • smoothness of f⁢t f t , where smoothness can be interpreted as the decay rate of | c n | c n .
  4. We can approximate a function by re-synthesizing using only some of the Fourier coefficients (truncating the F.S.)
    f N ′t=∑ n n≤|N| c n ⁢ei⁢ ω 0 ⁢n⁢t f N t n n N c n ω 0 n t
    (4)
    This approximation works well where f⁢t f t is continuous, but not so well where f⁢t f t is discontinuous. This idea is explained by Gibb's Phenomena.

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