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Fourier Transform and Hilber transform, properties

  1. Mar 5, 2015 #1
    Textbook says, Fourier transform expresses a function in time domain as a function in frequency domain. Basically, fourier transform gives two different expressions in terms of t domain and f domain but they represent the same signal.

    It also says Hilbert transform is a different type of transform because it gives a new signal not equal to the previous one.

    However, I do not think Fourier transform does 1 to 1 mapping since we have to integrate t for -infinity to +infinity. (ex, t=1 converts to f=constant.) I understand F.T shifts the domain, but strictly speaking, I don't think they have exactly the same structure (function-wise).

    In contrast, Hilber transform does 1 to 1 mapping as it shifts the original function by 90 degree. In this case function structure is conserved.


    I wonder why we say Hilbert transform is a totally different signal from the original one. is it because they are physically different?
     
  2. jcsd
  3. Mar 5, 2015 #2

    mathman

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    Fourier transform is 1-1. The inverse transform will give you the same function back (except on a set of measure 0).
     
  4. Mar 10, 2015 #3

    micromass

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    With what domain and codomain?
     
  5. Mar 10, 2015 #4

    mathman

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    Reals and reals.
     
  6. Mar 10, 2015 #5

    micromass

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    So the Fourier transform of a real number is a real number?
     
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