Question

A stable LTI system characterized by the differential equation $\frac{d^{2}y\left(t\right)}{d{t}^2}+\frac{4dy\left(t\right)}{dt}+3y\left(t\right)=\frac{dx\left(t\right)}{dt}+2x\left(t\right)$
the value of h(0) is_______ where h(t) is the impulse response of the given system.

• A. 0.5

Answer 1

The correct option is A 0.5
$\left(Assumeu\right(t){|}_{t=0}=0.5$
taking Fourier transform on both sides

$(j\omega {)}^2+4j\omega Y(\omega )+3Y(\omega )=j\omega X(\omega )+2X(\omega )$

$Y\left(\omega \right)\left[\right(j\omega {)}^2+4j\omega +3]=X(\omega \left)\right(2+j\omega )$

$H\left(\omega \right)=\frac{Y\left(\omega \right)}{X\left(\omega \right)}=\frac{2+j\omega }{(j\omega {)}^2+4(j\omega )+3}=\frac{(2+j\omega )}{(1+j\omega )(3+j\omega )}$

$H\left(\omega \right)=\frac{1}{2}\frac{1}{(1+j\omega )}+\frac{1}{2(3+j\omega )}$

taking inverse Fourier transform
$h\left(t\right)=\frac{1}{2}{e}^{-t}\left(t\right)+\frac{1}{2}{e}^{-3t}u\left(t\right)$

$h\left(0\right)=\frac{1}{2}u\left(0\right)+\frac{1}{2}u\left(0\right)=0.5$