Question

The response $h\left(t\right)$ of a linear time invariant system to an impulse $\delta \left(t\right)$, under initially relaxed condition is $h\left(t\right)=e^{-t}+e^{-2t}$. The response of this system for a unit step input u(t) is

• A. $u\left(t\right)+e^{-t}+e^{-2t}$
• B. ($e^{-t}+e^{-2t}\left)u\right(t)$
• C. $(1.5-e^{-t}-0.5e^{-2t})u\left(t\right)$
• D. $e^{-1}\delta \left(t\right)+e^{-2t}u\left(t\right)$

Answer 1

The correct option is C $(1.5-e^{-t}-0.5e^{-2t})u\left(t\right)$

Transfer function of system is impulse response of the system with zero initial conditions.

Transfer function =
$H\left(S\right)=L(e^{-t}+e^{-2t})$

$=\frac{1}{s+1}+\frac{1}{s+2}$

$H\left(s\right)=\frac{C\left(s\right)}{R\left(s\right)}=(\frac{1}{s+1}+\frac{1}{s+2})$

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

$C\left(s\right)=R\left(s\right).H\left(s\right)=\frac{1}{s}(\frac{1}{s+1}+\frac{1}{s+2})$

$=\frac{1}{s(s+1)}+\frac{1}{s(s+2)}$

$C\left(s\right)=(\frac{1}{s}-\frac{1}{s+1})+\frac{1}{2}(\frac{1}{s}-\frac{1}{s+2})$

$=\frac{1.5}{s}-\frac{1}{s+1}-\frac{0.5}{s+2}$

$Response=c\left(t\right)=L^{-1}\left[C\right(s\left)\right]$

$=L^{-1}[\frac{1.5}{s}-\frac{1}{s+1}-\frac{0.5}{s+2}]$

$c\left(t\right)=(1.5-e^{-t}-0.5e^{-2t})u\left(t\right)$