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.
Best Answer
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.