# Fourier Transform of $\frac{sin^2(3\pi C t) }{ 3 \pi^2 C t^2 }$?

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.

• 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). – Shahid M Shah Jun 19 '16 at 11:34
• @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? – JosephL Jun 19 '16 at 16:47
• I exactly meant that, the fourier transform of the required function is convolution of similar rect functions in frequency domains. – Shahid M Shah Jun 19 '16 at 17:03