Question

Consider an LTI system with impulse response $h\left(t\right)=e^{-5t}u\left(t\right)$. If the output of the system is $y\left(t\right)=e^{-3t}u\left(t\right)-e^{-5t}u\left(t\right)$ then the input, x(t) is given by

• A.
  $2e^{-3t}$ u(t)
• B.
  $2e^{-5t}$ u(t)
• C.
  $e^{-3t}$ u(t)
• D.
  $e^{-5t}$ u(t)

Answer 1

The correct option is A

$2e^{-3t}$ u(t)
$\text{Impulse response of an LTI system}$
$\text{= transfer function}$ $=\frac{Y\left(s\right)}{X\left(s\right)}=H\left(s\right)$

where, $y\left(t\right)=e^{-3t}u\left(t\right)-e^{-5t}u\left(t\right)$

$\therefore$ $Y\left(s\right)=\frac{1}{(s+3)}-\frac{1}{(s+5)}$

$=\frac{2}{(s+3)(s+5)}$

Also, $h\left(t\right)=e^{-5t}u\left(t\right)$

$\therefore$ $H\left(s\right)=\frac{1}{(s+5)}$

Therefore, $X\left(s\right)=\frac{Y\left(s\right)}{X\left(s\right)}$

$=\frac{2}{(s+3)(s+5)}\times (s+5)$

$=\frac{2}{(s+3)}$

$\therefore$ $\text{Input = x(t) = 2e}{}^{-3t}\text{u(t)}$