1. Impulse Response h(t)
Determines how the system reacts to an impulse.
2. Absolute Integrability ∫ |h(t)| dt
Total ∫ |h(t)| dt = Diverges logarithmically (∞)
3. Causality Support Timeline
If h(t) is non-zero anywhere in the red region (t < 0), the system anticipates the future.
4. Prediction Verdict
Is this system Causal?
Is this system BIBO Stable?
Correct! Non-causal (extends to -∞). NOT BIBO stable because the absolute integral of sinc diverges.
Common Conceptual Pitfalls
- "Bounded h(t) is enough for stability." False. A step function u(t) is strictly bounded between 0 and 1, but its absolute integral is infinite. If you feed a constant input of 1 into an integrator, the output ramps to infinity.
- "Just check if h(t) stays finite over time." BIBO stability strictly requires absolute integrability: ∫|h(t)|dt < ∞. Even if a signal decays (like sinc(t)), if it doesn't decay fast enough, the area diverges and the system is unstable.
- "Causal means no impulse response before t=0." Correct. If h(t) ≠ 0 for any t < 0, the system responds to an impulse before the impulse even occurs. This is physically impossible for real-time systems.