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If all the poles of H(z) are inside 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 mentioned
Correct answer is option 'C'. Can you explain this answer?
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If all the poles of H(z) are inside the unit circle, then the system i...
Explanation: If all the poles of H(z) are inside an unit circle, then it follows the condition that |z|>r < 1, it means that the system is both causal and BIBO stable.
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If all the poles of H(z) are inside the unit circle, then the system i...
Understanding System Stability and Causality
In control systems and signal processing, system stability and causality are critical concepts. When analyzing a discrete-time system characterized by its transfer function \( H(z) \), the location of the poles in the z-plane provides essential insights into these properties.
Poles Inside the Unit Circle
- A pole of \( H(z) \) is a value of \( z \) that makes the denominator zero.
- If all poles are located inside the unit circle (i.e., their magnitudes are less than 1), the system exhibits certain desirable properties.
BIBO Stability
- Bounded Input, Bounded Output (BIBO) Stability: A system is BIBO stable if every bounded input produces a bounded output.
- When all poles are inside the unit circle, the system is guaranteed to be BIBO stable. This is because the contributions of the system's impulse response decay to zero over time.
Causality
- A system is considered causal if the output at any time depends only on current and past inputs, not future inputs.
- For a causal system, the impulse response must be zero for negative time, which is typically the case for systems with poles inside the unit circle.
Conclusion: BIBO Stable and Causal
- When all poles of \( H(z) \) lie inside the unit circle, the system is both BIBO stable and causal.
- This ensures that the system is not only stable under bounded inputs but also adheres to the principle of causality, making it suitable for real-time applications.
Thus, the correct answer is option 'C': the system is BIBO stable and causal.
In control systems and signal processing, system stability and causality are critical concepts. When analyzing a discrete-time system characterized by its transfer function \( H(z) \), the location of the poles in the z-plane provides essential insights into these properties.
Poles Inside the Unit Circle
- A pole of \( H(z) \) is a value of \( z \) that makes the denominator zero.
- If all poles are located inside the unit circle (i.e., their magnitudes are less than 1), the system exhibits certain desirable properties.
BIBO Stability
- Bounded Input, Bounded Output (BIBO) Stability: A system is BIBO stable if every bounded input produces a bounded output.
- When all poles are inside the unit circle, the system is guaranteed to be BIBO stable. This is because the contributions of the system's impulse response decay to zero over time.
Causality
- A system is considered causal if the output at any time depends only on current and past inputs, not future inputs.
- For a causal system, the impulse response must be zero for negative time, which is typically the case for systems with poles inside the unit circle.
Conclusion: BIBO Stable and Causal
- When all poles of \( H(z) \) lie inside the unit circle, the system is both BIBO stable and causal.
- This ensures that the system is not only stable under bounded inputs but also adheres to the principle of causality, making it suitable for real-time applications.
Thus, the correct answer is option 'C': the system is BIBO stable and causal.
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