Question

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

(a) \(\frac{4z}{4z-1}\), |Z|>\(\frac{1}{4}\)

(b) \(\frac{4z}{4z-1}\), |Z|<\(\frac{1}{4}\)

(c) \(\frac{1}{1-4z}\), |Z|>\(\frac{1}{4}\)

(d) \(\frac{1}{1-4z}\), |Z|<\(\frac{1}{4}\)

The question was posed to me in an interview for job.

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

Answer 1

Right answer is (d) \(\frac{1}{1-4z}\), |Z|<\(\frac{1}{4}\)

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

= \(∑_{n=0}^∞ (4z)^n \)

= \(\frac{1}{1-4z}\), |Z|<\(\frac{1}{4}\).