Question

The value of z(a^k u[-k]) is _______________

(a) \(\frac{z}{z-a}\)

(b) \(\frac{z}{a-z}\)

(c) \(\frac{z^2}{z-a}\)

(d) \(\frac{a}{a-z}\)

The question was posed to me in an online quiz.

The question is from Properties of ROC topic in portion Laplace Transform and System Design of Signals and Systems

Answer 1

Right choice is (b) \(\frac{z}{a-z}\)

Easy explanation: z (a^k u [-k]) = \(∑_{k=-∞}^{-1} a^k z^{-k}\)

= \(∑_{m=1}^∞ a^{-m} z^m\)

= \(∑_{m=1}^∞ (a^{-1} z)^m \)

= \(\frac{a^{-1} z}{1-a^{-1} z} = \frac{z}{a-z}\).