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Question

# A discrete-time LTI system with the property such that if the input is x(n)=(12)nu(n)−14(12)n−1u(n−1) and output y(n)=(13)nu(n) The impulse response of the system is

A
2(13)nu(n)+3(14)nu(n)
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B
2(13)nu(n)+3(12)nu(n)
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C
2(13)nu(n)+3(12)nu(n)
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D
2(13)nu(n)+3(14)nu(n)
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Solution

## The correct option is D −2(13)nu(n)+3(14)nu(n)Taking DTFT of x(n) we get X(ejω)=11−12e−jω−1411−12e−jωe−jω=1−14e−jω1−12e−jω Y(ejω)=11−13e−jω Y(ejω)X(ejω)=H(ejω)=1−12e−jω(1−13e−jω)(1−14e−jω) H(ejω)=A1−13e−jω+B1−14e−jω A=(1−13e−jω)H(ejω)|e−jω=3=−2 B=(1−14e−jω)H(ejω)|e−jω=4=3 H(ejω)=−21−13e−jω+31−14e−jω h(n)=−2(13)nu(n)+3(14)nu(n)

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