Consider a continuous$-$time linear time$-$invariant $($LTI$)$ system.

$-$ When the input to this system is $()=!1$#$4\%(4),$ the output is $().$

$-$ When the input is $5-($#$)$

$5$# $,$ the output is $2()+!$# $()$

Find the impulse response $()$ of this system. $\{$Ans: $()=!($#$!.)(+4)\}$

Solution:

Note that

$$

$()=2!1$#$4\%(4)+!6(4)=2()+!6(4)$

The corresponding output is stated to be

$2()+!$# $()$

Therefore, the input $!6(4)$ yields the output $!$# $().$ Using the properties of LTI system,

we can say that

$-($homogeneous property$)$ the input $(4)$ gives $7!"8($#$)$

$7!$# $=!$#$46()$

$-($Time invariant$)$ the input $()=$G$4(4)$H gives $!($#$4")46(+4)=$

$!($#$!.)(+4)$

Thus the impulse response of the system is

$()=!($#$!.)(+4)$