# Find the Z-transform of x(n) = ($\frac{1}{2}$)^n u(n)*($\frac{1}{4}$)^n u(n).

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Find the Z-transform of x(n) = ($\frac{1}{2}$)^n u(n)*($\frac{1}{4}$)^n u(n).

(a) $\frac{z}{z-(1/2)} \frac{z}{z-(1/4)}$

(b) $\frac{z}{z-(1/2)} + \frac{z}{z-(1/4)}$

(c) $\frac{z}{z+(1/2)} * \frac{z}{z-(1/4)}$

(d) $\frac{z}{z-(1/2)} – \frac{z}{z+(1/4)}$

I got this question in exam.

This interesting question is from Properties of Z-Transforms in chapter Z-Transform and Digital Filtering of Signals and Systems

by (42.1k points)
Right option is (a) $\frac{z}{z-(1/2)} \frac{z}{z-(1/4)}$

Easiest explanation: We know that a^n u(n) ↔ $\frac{z}{z-a}$

Let x1 (n)=($\frac{1}{2}$)^n u(n) and x2 (n) = ($\frac{1}{4}$)^n u(n)

∴X1 (z) = $\frac{z}{z-(1/2)}$ and X2 (z) = $\frac{z}{z-(1/4)}$

Given x(n) = x1 (n) * x2 (n)

The convolution property of Z-transform states that

x1 (n) * x2 (n) ↔ X1 (z) X2 (z)

∴Z[x(n)] = X(z) = Z[x1 (n)*x2 (n)] = X1 (z) X2 (z) = $\frac{z}{z-(1/2)} \frac{z}{z-(1/4)}$.

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