# 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 o

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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

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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