Question

Statement for Linked Answer Questions 4.27 and 4.28:
The impulse response h(t) of a linear time-invariant continuous time system is given by h(t) = exp (-2t) u(t), where u(t) denotes the unit step function.
The frequency response $H\left(\omega \right)$ of this system in terms of angular frequency $\omega$,is given by, $H\left(\omega \right)$=

• A. $\frac{1}{1+j2\omega }$
• B. $\frac{sin\left(\omega \right)}{\omega }$
• C. $\frac{j\omega }{2+j\omega }$
• D. $\frac{1}{2+j\omega }$

Answer 1

The correct option is D $\frac{1}{2+j\omega }$

$\text{h(t)= exp(-2t)u(t)}$

$H\left(\omega \right)={\int }_{-\infty }^{\infty }h\left(t\right)e^{-j\omega t}dt$

$={\int }_0^{\infty }{e}^{-2t}{e}^{-j\omega t}dt={\int }_0^{\infty }{e}^{-(2+j\omega )t}dt$

$=-\frac{1}{2+j\omega }.e^{-(2+j\omega )t}{|}_0^{\infty }$

$=\frac{1}{2+j\omega }$