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Math Help - Fourier Transformation, calculus computations

  1. #1
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    Fourier Transformation, calculus computations

    Hi! I hope this is the right place to post this

    I need to compute the Fourier transform of f(x,y) = sin(x) sin(y). I know that there's the hard way and a couple of easy ways to do this and I want to find one of the easy ways instead of actually solving integrals. I've looked on the internet and there are lots of examples of doing it the long way, but nothing jumped out at me as being easier. I want to find the easier way because I also need to find the Fourier Transform of f(t) = cos(16 pi t) cos(64 pi t), which I'm totally lost on. I know that 16 is 1/4 of 64, which will push the frequencies further apart. I (think atlease) can draw the output, but for actually solving, I'm not quite sure what to do.

    Any help would be excellent!
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  2. #2
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    Quote Originally Posted by joecoolish View Post
    Hi! I hope this is the right place to post this

    I need to compute the Fourier transform of f(x,y) = sin(x) sin(y). I know that there's the hard way and a couple of easy ways to do this and I want to find one of the easy ways instead of actually solving integrals. I've looked on the internet and there are lots of examples of doing it the long way, but nothing jumped out at me as being easier. I want to find the easier way because I also need to find the Fourier Transform of f(t) = cos(16 pi t) cos(64 pi t), which I'm totally lost on. I know that 16 is 1/4 of 64, which will push the frequencies further apart. I (think atlease) can draw the output, but for actually solving, I'm not quite sure what to do.

    Any help would be excellent!
    For the first one are you transforming with respect to x,y or both?

    For the 2nd one I would recommend using the product to sum identity. This gives

    \displaystyle \cos(16 \pi t)\cos(64 \pi t) = \frac{\cos(16\pi t -64 \pi t)+\cos(16 \pi t +64\pi t)}{2}


    My 2nd question what do you mean by easier? Do you mean put them into a form and then look them up in a transform table?

    Since the above functions are not absolutely integrable we need to interpret their Fourier transforms as distributions not functions.
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  3. #3
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    Sorry I haven't gotten back. I've been away from my computer all week.

    Anyways, what I'm trying to do is get F(u,v) from f(x,y) where F(u,v) is the Fourier equivilant of f(x,y) in the space time domain (if that makes any sense).
    Last edited by joecoolish; March 21st 2011 at 01:27 PM.
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  4. #4
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    Ok, so I got the answers.

    The first one is F(u, v) = { .5i [Delta(u + 1/2pi) – Delta(u + 1/2pi)] * .5i [Delta(v + 1/2pi) – Delta(v + 1/2pi) ] }
    and the second one is F(u) = { 1/2 [Delta(u + 8) + Delta(u + 8)] * 1/2 [Delta(u + 32) + Delta(u + 32) ] }

    This is because sin(2pi s t) in the spatial domain = .5i [Delta(u + s) – Delta(u + s)] in the frequency domain, and cos = 1/2 [Delta(u + s) + Delta(u + s)]

    From there the solution was straight forward.

    Hope this helps someone!
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