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The impulse response of an LTI system which is continuous is H (t) = e−|t|. The system is ___________

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Answer & Explanation

Correct AnswerC. Stable but not causal

Quick Explanation

h(t)=e^{-|t|} is nonzero for t<0, so the system is not causal. But ∫|h(t)|dt over all time is finite (=2), so the LTI system is BIBO stable. Therefore it is stable but not causal.

Formula / Key Relation

Causality: h(t)=0 for t<0

Stability:

∫_{−∞}^{∞}|h(t)|dt

= 2∫₀^∞e^(−t)dt

= 2 < ∞

Detailed Explanation

Concept & Reasoning

For a continuous-time LTI system, causality requires h(t)=0 for t<0. BIBO stability requires absolute integrability. Here e^{-|t|} is symmetric and decays exponentially on both sides, making its absolute integral finite, but its negative-time portion violates causality.

How to Solve It in the Exam

Check causality and stability independently.

Important Exam Point

Two-sided decaying impulse responses are often stable but noncausal.

Related Revision Path

This question sits in the revision path Signals and Systems → LTI Systems → Causality and Stability .

Impulse Response Tests explanatory diagram

Quick Trick

Check causality and stability independently.

Common Mistake

Assuming exponential decay automatically implies causality.

Exam Tip

Two-sided decaying impulse responses are often stable but noncausal.

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