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Short Time Fourier Transform

  1. Oct 12, 2011 #1
    http://en.wikipedia.org/wiki/Square-integrable_function


    According to the tutorial: it says
    g*(x) is the complex conjugate of g

    but I can't get the idea from where this g(x) function comes, than why is it the complex conjugate?

    And it seems i can't visualize the inner product space? Some practical example would help me a lot.

    Thanks!
     
  2. jcsd
  3. Oct 12, 2011 #2

    olivermsun

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    The idea of square integrable functions is that the integral of the squared magnitude converges. For complex valued functions, |f(x)|^2 = ∫ f(x) f*(x) dx, which suggests a natural way to define both the "norm" and the "product" in the space of square integrable functions. You just say that the inner product <f, g> has to satisfy the property that |f|^2 = <f, f> and therefore <f, g> = ∫ f(x) g*(x) dx.
     
    Last edited: Oct 12, 2011
  4. Oct 12, 2011 #3

    HallsofIvy

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    In particular, you want [itex]|f|= <f , f>[/itex]. Since the "norm" is defined as [itex]\int f(x)f^*(x)dx= <f,f>[/itex] the natural way to define the "inner product" of two such functions, f and g, is [itex]<f, g>= \int f(x)g^*(x)dx[/itex].
     
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