Question

Find the Z-transform of x(n) = 2^n u(n-2).

(a) \(\frac{z}{z-2}\)

(b) \(\frac{z}{z+2}\)

(c) \(\frac{z}{z(z-2)}\)

(d) \(\frac{4}{z(z-2)}\)

This question was addressed to me by my school principal while I was bunking the class.

My query is from Properties of Z-Transforms in section Z-Transform and Digital Filtering of Signals and Systems

Answer 1

Correct option is (d) \(\frac{4}{z(z-2)}\)

For explanation I would say: Given x(n) = 2^n u(n-2)

Time shifting property of Z-transform states that

If x(n) ↔ X(z), then x(n-m) ↔ z^-m X(z)

Z[u(n-2)] = z^-2 Z[u(n)] = \(z^{-2} \frac{z}{z-1} = \frac{1}{z(z-1)}\)

The multiplication by an exponential sequence property of Z-transform states that

If x(n) ↔ X(z), then a^n x(n) ↔ X(z/a)

Z[2^n u(n-2)] = Z[u(n-2)]|z=(z/2) = \(\Big[\frac{1}{z(z-1)}\Big]_{z=(z/2)}\)

\( = \frac{1}{(z/2)[(z/2)-1]} = \frac{4}{z(z-2)}\).