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Question

A stable LTI system characterized by the differential equation d2y(t)dt2+4dy(t)dt+3y(t)=dx(t)dt+2x(t)
the value of h(0) is_______ where h(t) is the impulse response of the given system.

  1. 0.5

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Solution

The correct option is A 0.5
(Assume u(t)|t=0=0.5
taking Fourier transform on both sides

(jω)2+4 jω Y(ω)+3Y(ω)=jωX(ω)+2 X(ω)

Y(ω)[(jω)2+4 jω+3]=X(ω) (2+jω)

H(ω)=Y(ω)X(ω)=2+jω(jω)2+4(jω)+3=(2+jω)(1+jω)(3+jω)

H(ω)=121(1+jω)+12(3+jω)

taking inverse Fourier transform
h(t)=12et(t)+12e3tu(t)

h(0)=12u(0)+12u(0)=0.5

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