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Hann window and FFT

  1. Apr 10, 2008 #1
    Hi all,

    I am currently looking into the energy correction factor of the Hann window.

    So far I have found that to correct for the application of Hann-window and Fourier transform the result needs to be multiplied by sqrt(32)/sqrt(3).

    Can any of you explain how to get that factor ?

    The amplitude correction factor is 2, that much i grasp...

    There is a another factor of 2 as the FFT is performed as a complexed valued operation.

    These two factors multiplied is 4 (duhh !)

    And when removing that factor from sqrt(32)/sqrt(3) the result is ;

    4*sqrt(2)/sqrt(3).

    the sqrt(2)/sqrt(3) is the mysteries part. Can anyone explain how to reach this factor ?


    Best regards

    Lindemose
     
  2. jcsd
  3. Apr 10, 2008 #2

    rbj

    User Avatar

    dunno if my markup works. lessee...

    i guess it does. the definition for the Hann window (centered at zero and normalized in length) is:

    [tex] w(x) = \begin{cases}
    \frac{1}{2}\left(1 + \cos(\pi x) \right) & \mbox{if } |x| \le 1 \\[3pt]
    0 & \mbox{if } |x| > 1
    \end{cases} [/tex]

    now, compared to a no window (where w(x) is 1 for the same stretch of data), this window would reduce your mean voltage magnitude (or whatever signal) by a factor of 1/2. but if the issue is power or energy, you have to square it before integrating.

    [tex] \int_{-1}^{+1} w^2(x) \ dx = \int_{-1}^{+1} \frac{1}{4}\left(1 + \cos(\pi x) \right)^2 \ dx = \frac{3}{4} [/tex]

    if you did it to no window, the integral would be 2. so energy is reduced by a factor of 3/8. now perhaps they are talking about the r.m.s. reduction, then i think the factor is [itex]\sqrt{3/8}[/itex]. that almost looks right, except for a factor of 2.

    i can see where the [itex]\sqrt{3}[/itex] comes from, but not the [itex]\sqrt{32}[/itex]. i think it should be [itex]\sqrt{8}[/itex].
     
    Last edited: Apr 10, 2008
  4. Apr 11, 2008 #3
    Now that makes all the sense in the world, thank yoy very much.

    By the way the sqrt(32)should be sqrt(8) as you point out. I included the the amplitude correction factor of 2 within the number.

    Thanks again and best regards !

    Lindemose
     
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