Question

The unit impulse response of a linear time invarient system is the unit step function $u\left(t\right)$. For $t>0$, the response of the system to an excitation $e^{-at}u\left(t\right),a>0$ will be

• A. $a{e}^{-at}$
• B. $\left(1/a\right)(1-e^{-at})$
• C. $a(1-e^{-at})$
• D. $1-e^{-at}$

Answer 1

The correct option is B $\left(1/a\right)(1-e^{-at})$

$H\left(s\right)=\frac{1}{s}$
$R\left(s\right)=\frac{1}{s+a}$
$C\left(s\right)=R\left(s\right)H\left(s\right)=\frac{1}{s(s+a)}=\frac{1}{a}[\frac{1}{s}-\frac{1}{s+a}]$
$C\left(t\right)=\frac{1}{a}[1-e^{-at}]$