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Question

# The response h(t) of a linear time invariant system to an impulse δ(t), under initially relaxed condition is h(t)=e−t+e−2t. The response of this system for a unit step input u(t) is

A
u(t)+et+e2t
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B
(et+e2t)u(t)
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C
(1.5et0.5e2t)u(t)
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D
e1δ(t)+e2tu(t)
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Solution

## The correct option is C (1.5−e−t−0.5e−2t)u(t)Transfer function of system is impulse response of the system with zero initial conditions. Transfer function = H(S)=L(e−t+e−2t) =1s+1+1s+2 H(s)=C(s)R(s)=(1s+1+1s+2) R(s)=1s (Step input) C(s)=R(s).H(s)=1s(1s+1+1s+2) =1s(s+1)+1s(s+2) C(s)=(1s−1s+1)+12(1s−1s+2) =1.5s−1s+1−0.5s+2 Response=c(t)=L−1[C(s)] =L−1[1.5s−1s+1−0.5s+2] c(t)=(1.5−e−t−0.5e−2t)u(t)

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