Question

The z-transform of 3^n u[-n-1] is ___________

(a) \(\frac{z}{3-z}\), |Z|>3

(b) \(\frac{z}{3-z}\), |Z|<3

(c) \(\frac{3}{3-z}\), |Z|>3

(d) \(\frac{3}{3-z}\), |Z|<3

I got this question in my homework.

I'd like to ask this question from Properties of Z-Transforms in portion Z-Transform and Digital Filtering of Signals and Systems

Answer 1

Correct option is (b) \(\frac{z}{3-z}\), |Z|<3

The best I can explain: X (z) = \(∑_{n=-∞}^{-1} (3z^{-1})^n \)

= \(∑_{n=1}^∞ (z \frac{1}{3})^n \)

=  \(\frac{\frac{1}{3} z}{1-\frac{1}{3}z}\), |z|<3

= \(\frac{z}{3-z}\).