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Math Help - Help understanding a Fourier Transform question

  1. #1
    Super Member craig's Avatar
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    Help understanding a Fourier Transform question

    Calculate the Fourier Transform of g(x), where:

     g(x) =<br />
 \begin{cases}<br />
 x & 0 \le x \le 1 \\<br />
 2-x & 1 \le x \le 2 \\<br />
 0 & \text{otherwise}<br />
\end{cases}<br />

    The formula that I've got is the Fourier transform \hat{f}(s) of a function f(x) is a function defined by:

    \hat{f}(s) := \left\int^{\infty}_{-\infty}\right f(x) e^{-isx} dx

    My question is, how would I go about calculating the Fourier Transform for the above function? Which value of f(x) would I use, and which limits?

    Thanks in advance.
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  2. #2
    Super Member craig's Avatar
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    Just had a thought. Would I calculate the two integrals separately and then add them together?

    Also if this is the case, is this something you can always do?

    Thanks again
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by craig View Post
    Calculate the Fourier Transform of g(x), where:

     g(x) =<br />
 \begin{cases}<br />
 x & 0 \le x \le 1 \\<br />
 2-x & 1 \le x \le 2 \\<br />
 0 & \text{otherwise}<br />
\end{cases}<br />

    The formula that I've got is the Fourier transform \hat{f}(s) of a function f(x) is a function defined by:

    \hat{f}(s) := \left\int^{\infty}_{-\infty}\right f(x) e^{-isx} dx

    My question is, how would I go about calculating the Fourier Transform for the above function? Which value of f(x) would I use, and which limits?

    Thanks in advance.
    I'm pretty sure that it would be safe to say that \hat{g}(s) = \displaystyle\int_0^1 xe^{-isx}\,dx+\int_1^2(2-x)e^{-isx}\,dx due to the piecewise definition of g(x).

    I hope this helps!
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  4. #4
    Super Member craig's Avatar
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    Quote Originally Posted by Chris L T521 View Post
    I'm pretty sure that it would be safe to say that \hat{g}(s) = \displaystyle\int_0^1 xe^{-isx}\,dx+\int_1^2(2-x)e^{-isx}\,dx due to the piecewise definition of g(x).

    I hope this helps!
    Hi thanks for the reply.

    So I'm guessing that this would apply for any function defined like this?
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  5. #5
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by craig View Post
    Hi thanks for the reply.

    So I'm guessing that this would apply for any function defined like this?
    I want to say that the function should be piecewise continuous in order to do this; I'm pretty sure this is the case, but I'm not 100% sure.
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Chris L T521 View Post
    I want to say that the function should be piecewise continuous in order to do this; I'm pretty sure this is the case, but I'm not 100% sure.
    This is true for any integrable function.
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  7. #7
    Super Member craig's Avatar
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