Question

Match the following with respect to region of convergence

\( x(n) \)		ROC
A	Infinite duration causal sequence	I	Entire z-plane except at \( z = 0 \)
B	Finite duration causal sequence	II	Entire z-plane except at \( z = \infty \)
C	Infinite duration anticausal sequence	III	\( |z| > \alpha \), exterior of a circle of radius \( \alpha \)
D	Finite duration anticausal sequence	IV	\( |z| < \beta \), interior of a circle of radius \( \beta \)

• A. A -- II, B -- I, C -- III, D -- IV
• B. A -- IV, B -- III, C -- I, D -- II
• C. A -- III, B -- I, C -- IV, D -- II
• D. A -- I, B -- IV, C -- II, D -- III

Hint:

Causal (right-sided) \(\implies\) ROC is \(|z|>r_1\) (exterior).
Anticausal (left-sided) \(\implies\) ROC is \(|z|<r_2\) (interior).
Finite duration:
Causal (\(0 \le n \le N-1\)): ROC is \(|z|>0\) (all z except \(z=0\)).
Anticausal (\(-N+1 \le n \le 0\)): ROC is \(|z|<\infty\) (all z except \(z=\infty\)).
Two-sided finite (\(-N_1 \le n \le N_2\)): ROC is \(0<|z|<\infty\) (all z except \(z=0\) and \(z=\infty\)).

Answer 1

The Correct Option is C

A. Infinite duration causal sequence (\(x(n)=0\) for \(n<0\)): ROC is the exterior of a circle, \(|z|>\alpha\). Matches III.
B. Finite duration causal sequence (\(x(n) \neq 0\) for \(0 \le n \le N-1\)): Z-transform is \(\sum_{n=0}^{N-1} x(n)z^{-n}\). ROC is entire z-plane except possibly \(z=0\) (if \(x(n)\) has terms for \(n>0\)). Matches I.
C. Infinite duration anticausal sequence (\(x(n)=0\) for \(n>0\)): ROC is the interior of a circle, \(|z|<\beta\). Matches IV.
D. Finite duration anticausal sequence (\(x(n) \neq 0\) for \(-N+1 \le n \le 0\)): Z-transform is \(\sum_{n=-N+1}^{0} x(n)z^{-n}\). ROC is entire z-plane except possibly \(z=\infty\) (if \(x(n)\) has terms for \(n<0\)). Matches II. So, A-III, B-I, C-IV, D-II. \[ \boxed{\text{A -- III, B -- I, C -- IV, D -- II}} \]