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Math Help - Help!!! Fourier

  1. #1
    Newbie Quantum_Alpha's Avatar
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    Exclamation Help!!! Fourier

    My def. of FT is
    \int_{-\infty}^{\infty}f(t)e^{\frac{i2\pi t}{T}}dt, T= period

    * How to calculate Fourier transform for
    f1(t)= e^{-t}, for   t >= 0; and = e^t for t<0 (Which is the period here???)

    * f2(t)=2, for -1/2<t<1/2 , this gives me \frac{-e^{\alpha t}}{\alpha} |_{-\infty}^{\infty}=\infty I don't remember what to do with that...
    Maple says that FT in this case is 2\pi\delta(\pi).
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by Quantum_Alpha View Post
    My def. of FT is
    \int_{-\infty}^{\infty}f(t)e^{\frac{i2\pi t}{T}}dt, T= period

    * How to calculate Fourier transform for
    f1(t)= e^{-t}, for   t >= 0; and = e^t for t<0 (Which is the period here???)

    * f2(t)=2, for -1/2<t<1/2 , this gives me \frac{-e^{\alpha t}}{\alpha} |_{-\infty}^{\infty}=\infty I don't remember what to do with that...
    Maple says that FT in this case is 2\pi\delta(\pi).
    This gives me the impression that you are confusing Fourier series with Fourier transforms.

    If a function is periodic, with period T, then it has a Fourier series, in which the n-th (complex) Fourier coefficient is \hat{f}(n) = \frac1{T}\int_0^Tf(t)e^{i2\pi t/T}dt.

    Fourier transforms are for functions (non-periodic) that are integrable over the whole real line, and the Fourier transform is a function, defined by \hat{f}(w) = \int_{\infty}^{\infty}f(t)e^{-iwt}dt.

    For the two functions f_1 and f_2, the integrals for the Fourier transforms are
    \hat{f}_1(w) = \int_{-\infty}^0e^{t}e^{-iwt}dt + \int_0^{\infty}e^{-t}e^{-iwt}dt,
    \hat{f}_2(w) = 2\int_{-1/2}^{1/2}e^{-iwt}dt.
    Last edited by Opalg; December 22nd 2007 at 11:42 AM. Reason: correcting error in formula for Fourier coefficient
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  3. #3
    Newbie Quantum_Alpha's Avatar
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    Ok, I will resolve these integrals, I hope not to have problems with the \infty's
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  4. #4
    Newbie Quantum_Alpha's Avatar
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    Gives me \hat{f}_1(w)=\frac{2}{1-iw}, and \hat{f}_2(w) = -\frac{2}{iw}(e^{\frac{-iw}{2}}-e^{\frac{iw}{2}}).
    Are they ok?

    (Oh! About period, it was because in my def. I have w =2\pi/T)
    Last edited by Quantum_Alpha; December 21st 2007 at 05:27 PM.
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