Question

The z-transform of \((\frac{1}{4})^n\) (u[n] – u[n-5]) is __________

(a) \(\frac{z^5 – 0.25^5}{z^4 (z-0.25)}\), z>0.25

(b) \(\frac{z^5 – 0.25^5}{z^4 (z-0.25)}\), z>0

(c) \(\frac{z^5 – 0.25^5}{z^3 (z-0.25)}\), z<0.25

(d) \(\frac{z^5 – 0.25^5}{z^4 (z-0.25)}\), all z

I had been asked this question in an online quiz.

I'm obligated to ask this question of Properties of Z-Transforms in portion Z-Transform and Digital Filtering of Signals and Systems

Answer 1

The correct choice is (d) \(\frac{z^5 – 0.25^5}{z^4 (z-0.25)}\), all z

Explanation: X (z) = \(∑_{n=0}^4 (\frac{1}{4} z^{-1})^4 \)

= \(\frac{1-(\frac{1}{4} z^{-1})^5}{1-(\frac{1}{4} z^{-1})^1}\)

= \(\frac{z^5 – 0.25^5}{z^4 (z-0.25)}\), all z.