Question

If all the poles of \( H(z) \) are outside the unit circle, then the system is said to be:

• A. Only causal
• B. Only BIBO stable
• C. BIBO stable and causal
• D. None of the above

Hint:

For a discrete-time system to be BIBO stable, all poles must be inside the unit circle. If all poles are outside, the system is neither BIBO stable nor causal.

Answer 1

The Correct Option is D

Step 1: In Discrete-Time Systems, stability and causality are analyzed using the Z-transform. 
Step 2: The system is BIBO (Bounded-Input Bounded-Output) stable if: \[ \sum_{n=-\infty}^{\infty} |h(n)|<\infty \] 
Step 3: Stability Condition in the Z-Domain: 
- The system is BIBO stable if all poles lie inside the unit circle (\( |z|<1 \)). 
- If all poles are outside the unit circle, the system is not BIBO stable. 
Step 4: Causality Condition: 
- A system is causal if its Region of Convergence (ROC) is outside the outermost pole. 
- However, if all poles are outside the unit circle, the ROC is not valid for causality in practical systems. 
Step 5: Evaluating options: 
- (A) Incorrect: The system is not necessarily causal. 
- (B) Incorrect: The system is not BIBO stable. 
- (C) Incorrect: The system is neither BIBO stable nor causal. 
- (D) Correct: Since the system is neither BIBO stable nor causal, the correct choice is None of the above.