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Question

# The Fourier transform of h(n) is defined as H(ejω), where h(n) is the impulse response of the system whose input is x(n) and output y(n). If h(n)=3(12)nu(n)−2(−13)nu(n), then which of the following difference equations represent the system

A
y(n)=16y(n1)+16y(n2)+x(n)+2x(n1)
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B
y(n)=56y(n1)16y(n2)+x(n)
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C
y(n)=16y(n1)16y(n2)+2x(n1)
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D
y(n)=56y(n1)16y(n2)+x(n)
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Solution

## The correct option is A y(n)=16y(n−1)+16y(n−2)+x(n)+2x(n−1)h(n)=3(12)nu(n)−2(−13)nu(n) Taking 'z' transform, we get H(z)=3(1−12z−1)−2(1+13z−1) Y(z)X(z)=3+z−1−2+z−11−16z−1−16z−2=1+2z−11−16z−1−16z−2 Y(z)−16z−1Y(z)−16z−2Y(z)=X(z)+2z−1X(z) Taking inverse 'z' transform y(n)−16y(n−1)−16y(n−2)=x(n)+2x(n−1) y(n)=16y(n−1)+16y(n−2)+x(n)+2x(n−1)

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