Question

The ROC of u(n) = \(_k^nC\) is __________

(a) | z | > 1

(b) | z | < 1

(c) -1 < | z | < 1

(d) -1 < | z |

I have been asked this question in unit test.

I'd like to ask this question from Region of Convergence topic in division Laplace Transform and System Design of Signals and Systems

Answer 1

Right answer is (a) | z | > 1

For explanation I would say: Z [u (n)] = \(∑_{-∞}^∞\) \(_k^nC z^{-n}\)

= \(∑_{n=k}^∞\) \( _{k}^{n}C 2^n z^{-n}\)

\(∴ Z[u(n)] = ∑_{r=0}^∞\) \( ^{k+r}_{k}C z^{-(k+r)}\)

= \(z^{-k} ∑_{r=0}^∞\)\( _r^{k+r}C z^{-(r)}\)

= \(z^{-k} (1-1/z)^{-(k+1)}\)

The series is convergent for | 1/z | < 1 i.e., for | z | > 1

Hence, ROC is | z | > 1.