Question

The impulse response of an initially relaxed linear system is e${}^{-2t}$ u(t). To produce a response of $t.e^{-2t}$ u(t), the input must be equal to

• A.
  $\frac{1}{2}{e}^{-2t}$u(t)
• B.
  $e^{-t}$ u(t)
• C.
  $e^{-2t}$ u(t)
• D. $2e^{-t}$ u(t)

Answer 1

The correct option is C

$e^{-2t}$ u(t)
$H\left(S\right)=L\left[e^{-2t}u\right(t\left)\right]=\frac{1}{s+2}$

$Y\left(S\right)=L\left[t{e}^{-2t}u\right(t\left)\right]=\frac{1}{(s+2{)}^2}$

$\Rightarrow$ $X\left(S\right)=\frac{Y\left(s\right)}{X\left(s\right)}=\frac{1}{s+2}$

$\text{Hence, input x(t) = L}{}^{-1}\text{[X(s)] = e}{}^{-2t}\text{u(t)}$