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Math Help - Fourier transform - application, integral calculation

  1. #1
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    Fourier transform - application, integral calculation

    I'm trying to brush the dust of my transform tinkering. (Have to use it in some electronic stuff.) I just did an exercise where I calculate the F-transform of an even function.
    Let f(x) be:
    \displaystyle f(t)=\left\{\begin{matrix} 0 & x > 2\\ 2-x & 0 < x < 2 \end{matrix}\right.
    Find the even Fourier transform.

    I calculated the transform with this formula
    \displaystyle f(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt = \int_{-\infty}^{\infty} f(t) (cos(\omega t) + i sin(\omega t))dt

    Knowing that the function is even I write:
    \displaystyle f(\omega) = 2 \int_{0}^{\infty} (2-t) cos(\omega t) dt

    A bit integration gives, don't hesitate to call out errors.
    \displaystyle f(\omega) = \frac{4sin^{2}w}{w^2}

    Now as an application to this calculation I'm supposed to determine this integral.
    \displaystyle I = \int_0^{\infty} \frac{sin^{2}w}{w^2}

    This is where I like to get some help... I simply don't know how to do it. Calculating the integral, in an other way, I get \displaystyle \frac{\pi}{2}
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  2. #2
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    Re: Fourier transform - application, integral calculation

    Quote Originally Posted by liquidFuzz View Post
    I'm trying to brush the dust of my transform tinkering. (Have to use it in some electronic stuff.) I just did an exercise where I calculate the F-transform of an even function.
    Let f(x) be:
    \displaystyle f(t)=\left\{\begin{matrix} 0 & x > 2\\ 2-x & 0 < x < 2 \end{matrix}\right.
    Find the even Fourier transform.

    I calculated the transform with this formula
    \displaystyle f(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt = \int_{-\infty}^{\infty} f(t) (cos(\omega t) + i sin(\omega t))dt

    Knowing that the function is even I write:
    \displaystyle f(\omega) = 2 \int_{0}^{\infty} (2-t) cos(\omega t) dt

    A bit integration gives, don't hesitate to call out errors.
    \displaystyle f(\omega) = \frac{4sin^{2}w}{w^2}

    Now as an application to this calculation I'm supposed to determine this integral.
    \displaystyle I = \int_0^{\infty} \frac{sin^{2}w}{w^2}

    This is where I like to get some help... I simply don't know how to do it. Calculating the integral, in an other way, I get \displaystyle \frac{\pi}{2}
    The notation for the Fourier transform of a function should be different from the notation for the function itself. So you should write \hat{f}(\omega) rather than f(\omega).

    The Fourier inversion theorem says that you can retrieve f(t) from \hat{f}(\omega) by the formula f(t) = \frac1{2\pi}\int_{-\infty}^\infty \hat{f}(\omega)e^{it\omega}d\omega. Put t=0 in that formula and you will get the result you are looking for.
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    Re: Fourier transform - application, integral calculation

    Thanks! For slapping my fingers, and explaining how to do this.
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  4. #4
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    Re: Fourier transform - application, integral calculation

    How about this..?

    Assume that \hat{f}(w) converges to f(x), \hat{f}(w) = f(x), and that the Fourier transform is correct, \hat{f}(w) = \frac{4 sin^2 w}{w^2}

    Taking a arbitrary point on f(x)
    f(0) = 1 = \lim_{a \to \infty} \frac{1}{2 \pi} \int_0^a \frac{sin^2 x}{x^2} dx

    Gives
    1 = \frac{1}{2 \pi} \int_0^\infty \frac{4sin^2 x}{x^2} dx
    \drarrow \frac{\pi}{2} = \int_0^\infty \frac{sin^2 x}{x^2} dx
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