Question

An input $x\left(t\right)=exp(-2t)u\left(t\right)+\delta (t-6)$ is applied to an LTI system with impulse response h(t) = u(t). The output is

• A. $0.5[1-exp(-2t\left)\right]u\left(t\right)+u(t-6)$
• B. $0.5[1-exp(-2t\left)\right]u\left(t\right)+u(t+6)$
• C.
  $[1-exp(-2t\left)\right]u\left(t\right)+u(t+6)$
• D.
  $[1-exp(-2t\left)\right]u\left(t\right)+u(t-6)$

Answer 1

The correct option is A $0.5[1-exp(-2t\left)\right]u\left(t\right)+u(t-6)$

$x\left(t\right)=e^{-2t}u\left(t\right)+\delta (t-6)$

$X\left(s\right)=\frac{1}{s+2}+e^{-6s}$

$H\left(s\right)=\frac{1}{s}$

$\therefore {}^{.}Y\left(s\right)=X\left(s\right)\cdot H\left(s\right)$

$\Rightarrow Y\left(s\right)=\frac{1}{s(s+2)}+\frac{1}{s}{e}^{-6s}$

$\Rightarrow Y\left(s\right)=\frac{1}{2}(\frac{1}{s}-\frac{1}{s+2})+\frac{1}{s}{e}^{-6s}$

Taking inverse Laplace transform, we have
$y\left(t\right)=0.5(1-e^{-2t})u\left(t\right)+u(t-6)$