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Question

Consider a discrete time signal x[n] and its z-transform, X(z) given as

X(z)=z2+5zz22z3

If ROC of X(z) is |z|<1, then signal x[n] would be

A
[2(3)n(1)n]u[n]
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B
[2(3)n+(1)n]u[n1]
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C
2(3)nu[n1](1)nu[n]
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D
[2(3)n+1]+u[n]
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Solution

The correct option is B [2(3)n+(1)n]u[n1]

X(z)=z2+5zz22z3=z(z+5)(z3)(z+1)

X(z)z=z+5(z3)(z+1)=2z31z+1

Thus, X(z)=2zz3zz+1
ROC : |z| < 1, which is not exterior of circle outside the outermost pole = 3

So, x[n] is anti-causal given as,

x[n]=[2(3)n+(1)n]u[n1]

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