# Fourier Transform of sinc function (confused about $\operatorname{sinc}(\pi x)$ and $\operatorname{sinc}(x)$)

I am confused about the fourier transform of the $\operatorname{sinc}$ function. First I don't know if

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

or

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

Also, is the Fourier transform of $\operatorname{sinc} (a \pi f)$

$$\left( \frac{1}{\pi |a|} \right ) \text{rect} \left( \frac{f}{a \pi} \right )$$

or

$$\left( \frac{1}{a} \right) \text{rect} \left( \frac{f}{a} \right)$$?

Could someone help me understand which equation I need to use.

• $\text{sinc}_{n}(x) = \sin( \pi x)/(\pi x)$ is called the normalised sinc function. If you start with the un-normalised sinc function $$\text{sinc}_{u}(x) = \sin(x)/x$$ then mapping $x \mapsto \pi x \implies \text{sinc}_{u}( \pi x) = \sin( \pi x)/(\pi x) = \text{sinc}_{n}(x)$ gives the normalised sinc function. In response to your question about the Fourier transform, which convention of the FT are you using? – Mattos Sep 17 '16 at 3:12

This is one of those times where there are different definitions of what is meant by $\operatorname{sinc}(x)$ used by different people, analogous to mathematicians using $i=\sqrt{-1}$ and electrical engineers using $j=\sqrt{-1}$. When mathematicians say $\operatorname{sinc}(x)$ they usually mean $\sin(x)/x$. When information or signal processing people say it, they usually mean $\sin(\pi x)/(\pi x)$. So you're going to need to check which context you're working in, be explicit about what you mean, and ask when you're unsure.