GATE ECE 2011 | Question: 18

Question

If the unit step response of a network is $\left(1-e^{-\alpha t}\right)$, then its unit impulse response is

• A. $\alpha e^{-\alpha t}$
• B. $\alpha^{-1} e^{-\alpha t}$
• C. $\left(1-\alpha^{-1}\right) e^{-\alpha t}$
• D. $(1-\alpha) e^{-\alpha t}$

Comments

Option A

Answer 1

Since the unit step response of a network is given by the equation $1-e^{-αt}$, the unit impulse response of a network is the derivative of its unit step response. Therefore, we can find the unit impulse response by differentiating the given unit step response with respect to time (t).

Differentiating $1-e^{-αt}$ with respect to time (t):

$\frac{\mathrm{d} }{\mathrm{d} t}$ ($1-e^{-αt}$)

= 0 - $\frac{\mathrm{d} }{\mathrm{d} t}$ ($e^{-αt}$)

= 0 - (-α * $e^{-αt}$)

= α * $e^{-αt}$

Therefore, the correct answer to your question is Option A: α * $e^{-at}$