Question

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

Answer 1

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