Question

A signal is represented by
$x\left(t\right)=\{\begin{array}{c}1,|t|<1\\ 0,|t|>1\end{array}$
The Fourier transform of the convolved signal $y\left(t\right)=x\left(2t\right)\ast x\left(\frac{t}{2}\right)$ is

Answer 1

Given signal can be drawn as


For signal,

$F\left(\omega \right)$ is given by

Therefore,
$x\left(t\right)⟷X\left(\omega \right)=2Sa\left(\omega \right)$

Now, $x\left(t\right)⟷X\left(\omega \right)$

Then by time scaling,
$x\left(at\right)⟷\frac{1}{|a|}X\left(\omega /a\right)$

$\therefore x\left(2t\right)⟷Sa\left(\frac{\omega }{2}\right)....\left(i\right)$

and $x\left(\frac{t}{2}\right)⟷4Sa\left(2\omega \right)....\left(ii\right)$

Now, $y\left(t\right)=x\left(2t\right)\ast x\left(t/2\right)$

Convolution in time domain multiplication in frequency domain
$Y\left(\omega \right)=4Sa\left(\frac{\omega }{2}\right)Sa\left(2\omega \right)$

$Y\left(\omega \right)=\frac{4\sin \left(\frac{\omega }{2}\right)}{\frac{\omega }{2}}\frac{\sin \left(2\omega \right)}{2\omega }$

$Y\left(\omega \right)=\frac{4}{{\omega }^2}\sin \left(\frac{\omega }{2}\right)\sin \left(2\omega \right)$