Question

Let h(t) be the impulse response of a linear time invariant system. Then the response of the system for any input u(t) is

• A.
  $\frac{d}{dt}\underset{0}{\overset{t}{\int }}h\left(\tau \right)u(t-\tau )d\tau$
• B.
  $\underset{0}{\overset{t}{\int }}\left[\underset{0}{\overset{t}{\int }}h\right(\tau \left)u\right(t-\tau \left)d\tau \right]dt$
• C.
  $\underset{0}{\overset{t}{\int }}h\left(\tau \right)u(t-\tau )d\tau$
• D. $\underset{0}{\overset{t}{\int }}{h}^2\left(\tau \right)u(t-\tau )d\tau$

Answer 1

The correct option is C

$\underset{0}{\overset{t}{\int }}h\left(\tau \right)u(t-\tau )d\tau$
$\text{For LTI system}$

$\text{y(t) = u(t) * h(t)}$

$y\left(t\right)=\underset{-\infty }{\overset{\infty }{\int }}h\left(\tau \right)u(t-\tau )d\tau$

$\because u\left(t\right)=0fort<0$

$y\left(t\right)=\underset{0}{\overset{t}{\int }}h\left(\tau \right)u(t-\tau )d\tau$