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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(z12)(z14)
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[86(12)n+(14)n]u[n]
Y(z)=X(z).H(z)

The unit step input, X(z)=11z1 ( x[n]=unit step)

Y(z)=z3(z1)(z12)(z14)

Y(z)z=C1z1+C2z12+C3z14

C1=z2(z14)(z12)z=1=83

C2=z2(z1)(z14)z=12=2

C3=z2(z1)(z12)z=14=13

Y(z)=83zz12zz12+13zz14

taking inverse z-transform,

y[n]=13[86(12)n+(14)n]u[n]

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