I'm trying to find the Fourier Transform of the following rectangular pulse:
$$ x(t) = rect(t – 1/2) $$
This is simply a rectangular pulse stretching from 0 to 1 with an amplitude of 1. It is 0 elsewhere. I tried using the definition of the Fourier Tranform:
$$ X(\omega) = \int_0^1 (1)*e^{-j\omega*t}dt $$
However carrying out the relatively simple integration and subbing in the bounds results for me in this:
$$ X(\omega) = \frac{1}{j\omega}[e^{-j\omega} – 1] $$
& unfortunately wolfram alpha has a different answer when I use it to compute this fourier transform. It's got the sinc function;
I'd appreciate any help on this, if I've got some giant conceptual error. I have an exam on this stuff in a bit less than a week :/
Edit: also realized I used j; it's the same with i (the imaginary #)
Best Answer
They are equal to each other. Rewrite it using the following:
$$ ( e^{-j\omega} - 1 ) = e^{-j\omega/2}( e^{-j\omega/2} - e^{j\omega/2} ). $$