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Math Help - Fourier Transform of Hann Window

  1. #1
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    Fourier Transform of Hann Window

    Hi all,

    I'm trying to work out the Fourier transform for a Hann window , defined by f(x)=0.5(1-cos((2pi/T)x)), and really having no luck. Ultimately I want to find the transform of cos((2pi/T)x))f(x)... Any help would be very much appreciated- Thanks!
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  2. #2
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    Quote Originally Posted by malibupete
    Hi all,

    I'm trying to work out the Fourier transform for a Hann window , defined by f(x)=0.5(1-cos((2pi/T)x)), and really having no luck. Ultimately I want to find the transform of cos((2pi/T)x))f(x)... Any help would be very much appreciated- Thanks!
    First thing to sort out is the definition of the Hann window:

    <br />
f(x)=\left\{ \begin{array}{cc} 0.5(1-\cos(2\pi x /T)), & x \in [0,T]\\0, & \mbox{otherwise} \end{array} \right<br />

    This is the product of a rectangular window and a cosine function (plus a
    constant), and its FT is the convolution of the FT's of the rectangular
    window and the function (which is the sum of two delta functionals and so
    trivial to evaluate).

    RonL
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  3. #3
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    Ok, transforming the first part of the Hann window, I've got

    F(k)=0.5 (delta(k) + 0.5 (delta (k-2pi/T) + delta (k+2pi/T).

    and for the rectangular window I have

    W(k)= (sin(0.5kT))\k

    Am I right in thinking that the convolution of the two is

    S(k)= 0.5 ((sin(0.5kT))\k + 0.5 ((sin (0.5 (k-2pi/T))) + (sin (0.5 (k+2pi/T)))))?

    Would this then make the transform of the window multiplied by cos (2pi/T)x

    0.5(S(k-2pi/T) + (S(k+2pi/T))?

    Thanks ever so much for your help, and sorry about the mess- I've tried to make things as clear as possible, but I don't know how to input mathematical symbols on here...

    Thanks
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  4. #4
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    Quote Originally Posted by malibupete
    Ok, transforming the first part of the Hann window, I've got

    F(k)=0.5 (delta(k) + 0.5 (delta (k-2pi/T) + delta (k+2pi/T).

    and for the rectangular window I have

    W(k)= (sin(0.5kT))\k

    Am I right in thinking that the convolution of the two is

    S(k)= 0.5 ((sin(0.5kT))\k + 0.5 ((sin (0.5 (k-2pi/T))) + (sin (0.5 (k+2pi/T)))))?

    Would this then make the transform of the window multiplied by cos (2pi/T)x

    0.5(S(k-2pi/T) + (S(k+2pi/T))?

    Thanks ever so much for your help, and sorry about the mess- I've tried to make things as clear as possible, but I don't know how to input mathematical symbols on here...

    Thanks
    It looks OK on a quick scan, I will check it in detail when I have a bit more
    time.

    RonL
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  5. #5
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    Thanks

    Though I'm not sure it can be quite right- I should get two peaks and small values/0 everywhere else.. I can't see it cancelling down at the moment, but I'm probably missing something obvious. Either way, thanks for taking the time and effort to have a look.
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  6. #6
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    Quote Originally Posted by malibupete
    Thanks

    Though I'm not sure it can be quite right- I should get two peaks and small values/0 everywhere else.. I can't see it cancelling down at the moment, but I'm probably missing something obvious. Either way, thanks for taking the time and effort to have a look.
    No the peaks actualy overlap in such a way that with the aid of nearby
    zeros they merge into a single peak (I had a peek )

    RonL
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  7. #7
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    So am I right, then?
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  8. #8
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    No it doesn't mean I'm right, as I got several things wrong in my initial typing...

    W(k)=sin(0.5kT)/0.5k (Meaning that S(k) is different, as I forgot to put in the last two denominators, and obviously everything is 2x bigger...)
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