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I'm preparing for an exam in the signals and systems class I'm taking. One of the practice exams has a problem that requires you to take the Fourier transform of $\text{sinc}(4t)$.

From a table of Fourier transform pairs I found: $\dfrac{\omega_b}{\pi}\text{sinc}\left(\dfrac{\omega_b t}{\pi} \right)\Rightarrow \text{rect}(\omega/2\omega_b)$.

Using this I tried to match the given $\text{sinc}(4t)$ by rewriting it as $\dfrac14\dfrac{4\pi}{\pi}\text{sinc}\left(\dfrac{4\pi t}{\pi} \right)$. From this I get that $\omega_b = 4\pi$ and thus the Fourier transform should yield $\dfrac{1}{4}\text{rect}\left(\omega/8\pi \right)$.

But, in the exam solutions they show the Fourier transform to yield $\dfrac{\pi}{4}\text{rect}(\omega/8)$.

Any ideas where I'm going wrong?

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    You can't automatically use any table of Fourier transforms, since not everybody uses the same conventions (=definitions). Could you post the definition(s) you use in class and the ones that came with the table?2011-03-16
  • 0
    What exactly do you mean be conventions. As far as the transform pair goes the one I posted in the question is what is listed in our book.2011-03-16
  • 5
    There are two definitions of the sinc function in use; either $\sin x/x$ or $\sin \pi x/\pi x$. Could it be that the exam and the transform table differ in that regard?2011-03-16
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    Also, there are two normalizations for the Fourier Transform; $\int_{\mathbb{R}}f(t)e^{-2\pi ixt}\mathrm{d}t$ and $\int_{\mathbb{R}}f(t)e^{-ixt}\mathrm{d}t$.2012-05-08

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