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Question

# Consider the following table for an LTI system having an impulse response h(t). Input Output x(t) y(t) ddtx(t) −3y(t)+e−2tu(t) where, x(t)=2e−3tu(t−1) Then the impulse response h(t) of the system is

A
e32e2(t+1)u(t+1)
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B
e32e2(t+1)u(t+1)
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C
e32e2(t1)u(t1)
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D
e32e2(t+1)u(t+1)
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Solution

## The correct option is D e32e−2(t+1)u(t+1)System output is y(t) for the input x(t) ddt[x(t)]=ddt[2e−3tu(t−1)] =−3[2e−3tu(t−1)]+2e−3tδ(t−1) =−3x(t)+2e−3δ(t−1) Fortheinput−3x(t)+2e−3δ(t−1) Correspondingoutputofthesystemis−3y(t)+e−2tu(t) Herewecansay−3y(t)isoutputfortheinput−3x(t) and e−2tu(t) is output for input 2e−3δ(t−1) x1(t)=2e−3δ(t−1)correspondingoutput isy1(t)=e−2tu(t) ∴H(s)=Y1(s)X1(s)=1s+22e−3e−s H(s)=e32ess+2 h(t)=e32e−2(t+1)u(t+1)

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