How to obtain the CTFT of a sinc in terms of f in hertz (from the formula in terms of $ \omega $)
Recall:
$ x(t)=\frac{2 \sin \left( W t \right)}{\pi t } $
$ \mathcal{X}(\omega)=\left\{\begin{array}{ll}1, & \text{ if }|\omega| <W,\\ 0, & \text{else.}\end{array} \right. $
To obtain X(f), use the substitution
$ \omega= 2 \pi f $.
More specifically
$ X(f)=\mathcal{X}(2\pi f)=\left\{\begin{array}{ll}1, & \text{ if }|f| <\frac{W}{2\pi},\\ 0, & \text{else.}\end{array} \right. \ $
