BIBO Stability Explained with Examples
While BIBO(Bounded-Input, Bounded-Output) stability is often defined mathematically, it is best understood through examples.
Continuous-Time
Discrete-Time
It means that impulse response must be absolutely integrable or summable if input is bounded.
Example 1: Stable System
Impulse response
This decays over time, and the system is stable.
Example 2: Unstable System
Impulse response
This grows exponentially,and the system is unstable.
Interpretation
- Decaying impulse response โ stable
- Growing impulse responseย โ unstable
- โAll FIR filters are inherently BIBO stable, while IIR filters require stability verification.โ
Physical Meaning
- Stable systems dissipate energy
- Unstable systems amplify energy
- โAn unstable system amplifies energy uncontrollably over time.โ
Relation to Transfer Function (Digital Filter)
BIBO stability is determined by the denominator of the transfer function. Therefore, all poles are inside the unit circle
- |z| < 1ย โ decaying response (stable)
- |z| > 1 โ growing response (unstable)
Z-plane representation of the unit circle where each point ejฯ corresponds to a frequency component, with angle ฯ indicating the signal frequency.
Example: BIBO stable vs unstable
input x: sine
impulse response h1: pulse (finte duration)
impulse response h2: sine
BIBO stable system: h1 (pulse) is absolutely summable
BIBO unstable system: h2 (sine) is not absolutely summable
Conclusion
A system is BIBO stable if every bounded input produces a bounded output, which is guaranteed when its impulse response is absolutely summable.
Suggested Further Reading
BIBO Stability Explained with Examples
While BIBO(Bounded-Input, Bounded-Output) stability is often defined mathematically, it is best understood through examples.
Continuous-Time
Discrete-Time
It means that impulse response must be absolutely integrable or summable if input is bounded.
Example 1: Stable System
Impulse response
This decays over time, and the system is stable.
Example 2: Unstable System
Impulse response
This grows exponentially,and the system is unstable.
Interpretation
Physical Meaning
Relation to Transfer Function (Digital Filter)
BIBO stability is determined by the denominator of the transfer function. Therefore, all poles are inside the unit circle
Z-plane representation of the unit circle where each point ejฯ corresponds to a frequency component, with angle ฯ indicating the signal frequency.
Example: BIBO stable vs unstable
input x: sine
impulse response h1: pulse (finte duration)
impulse response h2: sine
Conclusion
A system is BIBO stable if every bounded input produces a bounded output, which is guaranteed when its impulse response is absolutely summable.
Suggested Further Reading