Question

A continuous, linear time-invariant filter has an impulse response h(t) described by
$h\left(t\right)=\{\begin{array}{cc}3& for0\le t\le 3\\ o& otherwise\end{array}$
When a constant input of value 5 is applied to this filter, the steady state output is ____

• A. 45

Answer 1

The correct option is A 45
$h\left(t\right)=h\left(t\right)=\{\begin{array}{cc}3& for0\le t\le 3\\ o& otherwise\end{array}$

$h\left(t\right)=3\left[u\right(t)-u(t-3\left)\right]$

$H\left(s\right)=3[\frac{1}{s}-\frac{e^{-3s}}{s}]$

$\therefore$ $x\left(t\right)=5$

Then $X\left(5\right)=\frac{5}{s}$

We know that
$Y\left(s\right)=H\left(s\right)X\left(s\right)=[\frac{1}{s}-\frac{e^{-3s}}{s}]$

Steady state output
$={lim}_{s\to 0}s.Y\left(s\right)={lim}_{s\to o}s.\frac{15}{s}\frac{1-e^{-3s}}{s}$

$=15{lim}_{s\to 0}3e^{-3s}$ [because Hospitals rule]

$=45$