Which of the following describes the impulse response of a stable, causal LTI system?

• A. It is absolutely integrable: ∫_{-∞}^{∞} |h(t)| dt < ∞
• B. It is non-causal
• C. It has finite energy but infinite L1 norm
• D. It grows without bound as t → ∞

Answer: (A)

The main idea is how BIBO stability constrains the impulse response of a stable, causal LTI system. For BIBO stability, the impulse response must be absolutely integrable, meaning the integral of its absolute value is finite. If the system is causal, h(t) = 0 for t < 0, so this reduces to ∫_0^∞ |h(t)| dt < ∞. Then, with any bounded input x(t) satisfying |x(t)| ≤ M, the output y(t) = (h * x)(t) satisfies |y(t)| ≤ M ∫_0^∞ |h(τ)| dτ. Because that integral is finite, the output remains bounded for all t, which is exactly the stability condition. The other statements don’t describe a stable, causal impulse response: non-causality contradicts the premise, finite energy with infinite L1 norm would not guarantee bounded output for all bounded inputs, and growth without bound would mean instability.