Question

The z-transform of \((\frac{2}{3})^{[n]}\) is ____________

(a) \(\frac{-5z}{(2z-3)(3z-2)}\), –\(\frac{3}{2} \) < z < –\(\frac{2}{3}\)

(b) \(\frac{-5z}{(2z-3)(3z-2)}\), \(\frac{2}{3}\) < |z| < \(\frac{3}{2} \)

(c) \(\frac{5z}{(2z-3)(3z-2)}\),  \(\frac{2}{3}\) < |z|

(d) \(\frac{5z}{(2z-3)(3z-2)}\), –\(\frac{3}{2} \) < z< –\(\frac{2}{3}\)

The question was asked in an interview for job.

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

Answer 1

Right answer is (b) \(\frac{-5z}{(2z-3)(3z-2)}\), \(\frac{2}{3}\) < |z| < \(\frac{3}{2} \)

Easiest explanation: X(z) = \(∑_{-∞}^{-1} (\frac{3}{2} z^{-1})^n  + ∑_{n=0}^∞ (\frac{2}{3} z^{-1})^n \)

= \(\frac{-1}{(1-\frac{3}{2} z^{-1})} + \frac{1}{(1-\frac{2}{3} z^{-1})}\)

= \(\frac{-5z}{(2z-3)(3z-2)}\), \(\frac{2}{3}\) < |z| < \(\frac{3}{2} \).