Question

The ROC of u(n) = 4^n, for n<0; 2^n, for n≥0 is ___________

(a) 0<z<1

(b) z<4

(c) 2<z

(d) 2<z<4

This question was addressed to me in final exam.

I would like to ask this question from Region of Convergence topic in chapter Laplace Transform and System Design of Signals and Systems

Answer 1

Correct choice is (d) 2<z<4

To elaborate: \(Z [u (n)] = ∑_{-∞}^∞ u(n)  Z^{-n}\)

= \(∑_{-∞}^{-1} 4^n  Z^{-n} + ∑_{n=0}^∞ 2^n  Z^{-n}\)

= \(\frac{z}{4} \Big\{1 + \frac{z^1}{4} + \frac{z^2}{4} +⋯….\Big\} + \Big\{1 + \frac{2}{z} + \frac{2^2}{z} + ….\Big\}\)

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

= \(\frac{z}{4-z} + \frac{z}{z-2}\)

= \(\frac{2z}{(4-z)(z-2)}\)

Now, the above series will be convergent if | \(\frac{z}{4}\) |<1 and | \(\frac{2}{z}\) |<1.

That is 2<z<4.