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Question

# Consider the system shown below, If x[n] is a discrete time signal given to a system with its impulse response as h[n] and its transformation H(z)=z2(z−12)(z−14) The step response of the system is

A
y[n]=[82(12)n+13(14)n]u[n]
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B
y[n]=[833(12)n+13(14)n]u[n]
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C
y[n]=13[86(12)n+(14)n]u[n]
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D
y[n]=[8323(12)n+(14)n]u[n]
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Solution

## The correct option is C y[n]=13[8−6(12)n+(14)n]u[n]Y(z)=X(z).H(z) The unit step input, X(z)=11−z−1 (∵ x[n]=unit step) Y(z)=z3(z−1)(z−12)(z−14) Y(z)z=C1z−1+C2z−12+C3z−14 C1=z2(z−14)(z−12)∣z=1=83 C2=z2(z−1)(z−14)∣z=12=−2 C3=z2(z−1)(z−12)∣z=14=13 Y(z)=83zz−1−2zz−12+13zz−14 taking inverse z-transform, y[n]=13[8−6(12)n+(14)n]u[n]

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