C is a positive constant. How would you calculate the Fourier Transform of $\frac{sin^2(3\pi C t) }{ 3 \pi^2 C t^2 }$? As it is not easy to calculate the Fourier integral $ \int_{-\infty}^{\infty} \frac{sin^2(3\pi C t) }{ 3 \pi^2 C t^2 } \cdot e^{-j 2 \pi f t} dt $ or look up in a Fourier table.

$\operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x}~.$

$\frac{sin^2(3\pi C t) }{ 3 \pi^2 C t^2 } = 3C sinc(3Ct)^2$

By using the signal processing definition of the Fourier transform, it should be something like this according the WolframAlpha.

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    $\begingroup$ We can write $\frac{\sin^2(3\pi Ct)}{3\pi^2Ct^2}$=$3C\left(\frac{\sin(3\pi Ct)}{3\pi Ct}\right)^2$=$3C sinc(3\pi Ct)^2$. Now we know fourier transform of sinc function. The fourier transform of product of sinc function will be convolution of fourier transform of sinc functions (which is rect function). $\endgroup$ – Shahid M Shah Jun 19 '16 at 11:34
  • $\begingroup$ @ShahidMShah In signal processing $\operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x}~.$ So probably you mean $3C sinc(3Ct)^2$? The Fourier transform of a sinc is a rect, but wouldn't multiplying two sinc functions in time domain, mean convolving two rects in frequency domain? $\endgroup$ – JosephL Jun 19 '16 at 16:47
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    $\begingroup$ I exactly meant that, the fourier transform of the required function is convolution of similar rect functions in frequency domains. $\endgroup$ – Shahid M Shah Jun 19 '16 at 17:03

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