I'm trying to find the input of an LTI system.
So far I'm at the point where I know
$$ X(f) = \frac{sin(2\pi f)}{\pi f}$$
I also know that a $$sinc(t) = \frac{sin(\pi f)}{\pi f}$$
And the Fourier transform table says that:
$$ x(t) = rect(\frac{t}{\tau}) <=> X(f) = \tau sinc(\tau f)$$
I having trouble getting my $X(f)$ into the proper form to be converted into the time domain.
My attempt is this:
$$ X(f) = \frac{sin(2\pi f)}{\pi f} * \frac{2}{2}$$
$$ X(f) = 2\frac{sin(2\pi f)}{2\pi f} $$
$$ X(f) = 2{sinc(2 f)} $$
$$ X(f) = \tau {sinc(\tau f)} $$
$$ X(t) = rect(\frac{t}{\tau}) $$
$$\tau = 2$$
Therefore: $$x(t) = rect(\frac{t}{2})$$
But I feel like I'm making a mistake. Any thoughts? Thanks.