Question

In the system shown below, $x\left(t\right)=\left(\sin t\right)u\left(t\right)$. In steady-state, the response $y\left(t\right)$ will be

• A. $\frac{1}{\sqrt{2}}\sin (t-\frac{\pi }{4})$
• B. $\frac{1}{\sqrt{2}}\sin (t+\frac{\pi }{4})$
• C. $\frac{1}{\sqrt{2}}{e}^{-t}\sin t$
• D. $\sin t-\cos t$

Answer 1

The correct option is A $\frac{1}{\sqrt{2}}\sin (t-\frac{\pi }{4})$

$y\left(t\right)=x\left(t\right)\ast h\left(t\right)$
$Y\left(s\right)=X\left(s\right).H\left(s\right)$
$H\left(j\omega \right)=\frac{1}{s+1}=\frac{1}{\sqrt{2}}\angle -{45}^{\circ }$
$=\frac{1}{\sqrt{2}}\angle -\frac{\pi }{4}$
$x\left(t\right)=\sin \left(t\right)u\left(t\right)$
$\therefore y\left(t\right)=\frac{1}{\sqrt{2}}\sin (t-\frac{\pi }{4})$