Question

If the response of LTI continuous time system to unit step input is \(\left( {\frac{1}{2} - \frac{1}{2}{e^{ - 2t}}} \right)\), then impulse response of the system is
1. \(\left( {\frac{1}{2} - \frac{1}{2}{e^{ - 2t}}} \right)\)
2. \(\left( {{e^{ - 2t}}} \right)\)
3. \(\left( {1 - {e^{ - 2t}}} \right)\)
4. Constant

Answer 1

Correct Answer - Option 2 : \(\left( {{e^{ - 2t}}} \right)\)

Concept:

A transfer function is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input by assuming initial conditions are zero.

TF = L[output]/L[input]

\(TF = \frac{{C\left( s \right)}}{{R\left( s \right)}}\)

For unit impulse input i.e. r(t) = δ(t)

⇒ R(s) = δ(s) = 1

Now transfer function = C(s)

Therefore, the transfer function is also known as the impulse response of the system.

Transfer function = L[IR]

IR = L-1 [TF]

Analyses:-

\(s\left( t \right) = \frac{1}{2} - \frac{1}{2}{e^{ - 2t}}\)

\(h\left( t \right) = \frac{{ds\left( t \right)}}{{dt}}\)

\(= \frac{d}{{dt}}\left( {\frac{1}{2} - \frac{1}{2}{e^{ - 2t}}} \right)\)

= e^-2t