Question

The value of z{[k-1] u(k)} is _______________

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

(b) \(\frac{2z-z^2}{(z-1)^2}\)

(c) \(\frac{z^2}{(z-1)^2}\)

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

This question was posed to me during an interview.

My enquiry is from Properties of ROC in division Laplace Transform and System Design of Signals and Systems

Answer 1

Right option is (b) \(\frac{2z-z^2}{(z-1)^2}\)

To explain: z {[k-1] u (k)} = z {k u (k) – u (k)}

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

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

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

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