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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)n1u(n1)

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)=1112ejω141112ejωejω=114ejω112ejω

Y(ejω)=1113ejω

Y(ejω)X(ejω)=H(ejω)=112ejω(113ejω)(114ejω)

H(ejω)=A113ejω+B114ejω

A=(113ejω)H(ejω)|ejω=3=2

B=(114ejω)H(ejω)|ejω=4=3

H(ejω)=2113ejω+3114ejω

h(n)=2(13)nu(n)+3(14)nu(n)

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