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Math Help - fourier integral

  1. #1
    Junior Member Pinsky's Avatar
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    fourier integral

    I'm having problems with understanding Fourier integral. I only have one example solved, but i don't have a clue what is done there.

    I have to transform this function into a Fourier integral

    f(x) = \left\{<br />
\begin{array}{c l}<br />
  A\cos{\omega_0 x},x\epsilon <-\frac{\pi}{2\omega_0},\frac{\pi}{2\omega_0}> \\<br />
  0 ,\mbox{ otherwise}<br />
\end{array}\right.<br />

    What next?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Pinsky View Post
    I'm having problems with understanding Fourier integral. I only have one example solved, but i don't have a clue what is done there.

    I have to transform this function into a Fourier integral

    f(x) = \left\{<br />
\begin{array}{c l}<br />
  A\cos{\omega_0 x},x\epsilon <-\frac{\pi}{2\omega_0},\frac{\pi}{2\omega_0}> \\<br />
  0 ,\mbox{ otherwise}<br />
\end{array}\right.<br />

    What next?

    Start with the definition of the Fourier Transform:

    \mathcal{F}f (\omega) = \int_{-\infty}^{\infty} f(x) e^{i\omega x}dx

    (your definition may vary as there is no standard about where the constants go in the FT IFT pair).

    Now your function is zero outside of the inteval \left[-\frac{\pi}{2\omega_0},\frac{\pi}{2\omega_0}\right], so the integral can be done over this interval:

    \mathcal{F}f (\omega) = \int_{-\frac{\pi}{2\omega_0}}^{\frac{\pi}{2\omega_0}} f(x) e^{i\omega x}dx=<br />
\int_{-\frac{\pi}{2\omega_0}}^{\frac{\pi}{2\omega_0}}  A\cos(\omega_0 x)\;e^{i\omega x}dx

    and I will leave the rest to you.

    RonL
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