Question

The transfer function of causal LTI system is H(s) = 1/s. If the input to the system is $x\left(t\right)=\left[sin\right(t\left)/\pi t\right]u\left(t\right)$ . where u(t) is a unit step function, the system output y(t) as $t\to \infty is$_____

Answer 1

$H\left(s\right)=\frac{1}{s}\Rightarrow Integrator$

$x\left(t\right)=\frac{sint}{\pi t}u\left(t\right)$

$So,y\left(t\right)={\int }_o^t\frac{sin\tau }{\pi \tau }d\tau$

$y(t{)}_{t\to \infty }=\frac{1}{\pi }{\int }_0^{\infty }\frac{sin\tau }{\tau }dr$

${\int }_0^{\infty }\frac{sin\tau }{\tau }d\tau =\frac{\pi }{2}$

So, $y(t{)}_{t\to \infty }=\frac{1}{\pi }\times \frac{\pi }{2}=\frac{1}{2}=0.5$