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Math Help - Fourier Transform of a Bessel Function

  1. #1
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    Fourier Transform of a Bessel Function

    Hi all,
    I'm working on a project (imaging) where I have a Bessel function that I need to take the Fourier Transform of. I looked the transform up in a table, and was given:

    W(q) = J1(2*pi*a*q)/(pi*a*q) <=> w(r/2a) = {1 if r < a, 0 if r > a}

    Where q is a radial distance in the starting plane, and r is the radial distance in the end frame. So, basically, the transform of a Bessel function of the first kind (n=1) is a window function. What I am confused about is the the parameter "a." I know that I can use it to vary the "width" of the Bessel function (another thing I am slightly confused about, how is the "width" of a Bessel function formally defined?), but what does w( r/2a ) mean? I would like to be able to predict for a given "a" the width of the window function produced from the above transform (or is it as simple as the window has width a?).

    Thanks for your consideration of this issue.

    Dillon
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by haystack View Post
    Hi all,
    I'm working on a project (imaging) where I have a Bessel function that I need to take the Fourier Transform of. I looked the transform up in a table, and was given:

    W(q) = J1(2*pi*a*q)/(pi*a*q) <=> w(r/2a) = {1 if r < a, 0 if r > a}

    Where q is a radial distance in the starting plane, and r is the radial distance in the end frame. So, basically, the transform of a Bessel function of the first kind (n=1) is a window function.
    That is not what you have W(q) is not a Bessel function so your w is not the FT of a Bessel function.

    (Also, without checking that does not look a plausible FT pair either, now I have checked it I can confirm that is not a FT pair)

    See the table of FT pairs here it should have what you want.

    RonL
    Last edited by CaptainBlack; June 26th 2008 at 10:02 PM.
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