Q$2($a$)$ Given two signals;

$x_1(t)=e^{-at}u(t),a>0,$ and $,x_2(t)=e^{-bt}u(t),b>0$

where $ab.$

If $x(t)$ is an output of a convolution process between $x_1(t)$ and $x_2(t),$ that is $x(t)=x_1(t)**x_2(t).$ Find:

$($i$)$ Fourier transform of $x(t).$

$($ii$)$ Signal $x(t)$ using inverse Fourier transform of signal in Q$2($a$)($i$).$

$($b$)$ A basic modulator circuit is shown in Figure Q$2\mathrm{.}1.$ Modulation is a multiplication between voltage signal, $v(t),$ and a carrier signal, $c(t).$ The process yields an output signal, $m(t).$

$($i$)$ Analyze the Fourier transform of output signal, $m(t)$ by using modulation properties.

$($ii$)$ Sketch the spectrum signal of $M().$

$3$

$($c$)$ Analyze the output voltage, $v_o(t)$ of the electrical circuit shown in Figure Q$2\mathrm{.}2$ for the input voltage

$v_i(t)=3e^{-5t}u(t)V$