Question

A time domain system is LTI and has impulse response of h(t) given by
$h\left(t\right)=\cos {\omega }_{0}t+j\sin {\omega }_{0}t$
Thus impulse response is complex in nature if input to the system is
x(t) = u(t+3) - u(t-3) then output is y(t), if y(0) = 6 then value of ${\omega }_0$ tends to _____.

• A. 0

Answer 1

The correct option is A 0
$h\left(t\right)=\cos {\omega }_{0}t+j\sin {\omega }_{0}t$

$\Rightarrow h\left(t\right)=e^{j{\omega }_{0}t}$

and x(t) = u(t-3) - u(t+3)

So, y(t) = x(t)*h(t)

$y\left(0\right)={\int }_{-\infty }^{\infty }h(t-\tau )x\left(\tau \right)d\tau ={\int }_{-3}^3{e}^{-j{\omega }_0\tau }d\tau$

$\Rightarrow y\left(0\right)=\frac{e^{-3j{\omega }_0}-e^{+3j{\omega }_0}}{-j{\omega }_0}$

$\Rightarrow y\left(0\right)=\frac{2\sin 3{\omega }_0}{{\omega }_0}$

and y(0) = 6 at $\omega \ne 0$