Question

Consider a LTI system whose input is x(t) and output is y(t). If y(t) and x(t) are related to each other by relation $y\left(t\right)={\int }_{-\infty }^{t-10}{e}^{-(t-5-\tau )}x\left(\tau \right)d\left(\tau \right)$ then impulse response of system will be

• A. $e^{(t-5)}u(t-10)$
• B. $e^{-(t-5)}u(t-5)$
• C. $e^{-(t-5)}u(t-10)$
• D. $e^{(t-5)}u(t-5)$

Answer 1

The correct option is C $e^{-(t-5)}u(t-10)$

Since $y\left(t\right)=x\left(t\right)\ast h\left(t\right)$

$y\left(t\right)={\int }_{-\infty }^{\infty }h(t-\tau )x\left(\tau \right)d\left(\tau \right)$

Given that
$y\left(t\right)={\int }_{-\infty }^{t-10}{e}^{-(t-5-\tau )}x\left(\tau \right)d\tau$

Comparing we get
$h(t-\tau )=e^{-(t-5-\tau )}for\tau <t-10$
So h ($-\tau$ ) = $e^{(5+\tau )}for(\tau <-10$)
Thus h ($\tau$ ) = $e^{(5-\tau )},for(\tau >10)$

$\Rightarrow h\left(t\right)=e^{-(t-5)}u(t-10)$