Question

If the unit step response of a network is \(\left( {1 - {e^{ - \alpha t}}} \right)\), then its unit impulse response will be
1. \(\alpha {e^{ - \alpha t}}\)
2. \(\alpha {e^{ - \frac{1}{\alpha }t}}\)
3. \(\frac{1}{\alpha }{e^{ - \alpha t}}\)
4. \(\left( {1 - \alpha } \right){e^{ - \alpha t}}\)

Answer 1

Correct Answer - Option 1 : \(\alpha {e^{ - \alpha t}}\)

The unit step response of network is

\(\begin{array}{l} c\left( t \right) = 1 - {e^{ - \alpha t}}\\ \frac{d}{{dt}}\left( {unit\;step\;response} \right) = impulse\;response\\ \frac{d}{{dt}}\left( {1 - {e^{ - \alpha t}}} \right) = \alpha {e^{ - \alpha t}} \end{array}\)