Which condition guarantees BIBO stability for an LTI system?

Which condition guarantees BIBO stability for an LTI system?

• A. The impulse response h[n] is absolutely summable, i.e., sum |h[n]| < ∞.
• B. The impulse response has finite length.
• C. The impulse response has no zeros.
• D. The impulse response is symmetric.

Answer: (A)

In a discrete-time LTI system, BIBO stability means that every bounded input produces a bounded output. This holds if the impulse response is absolutely summable: the sum over all n of |h[n]| is finite. Why this works: the output is the convolution y[n] = ∑ h[k] x[n−k]. If the input is bounded by some B, then |y[n]| ≤ B ∑ |h[k]|. For this to be true for every n, the sum ∑ |h[k]| must be finite. That absolute summability is both necessary and sufficient for BIBO stability. Finite-length impulse responses are a concrete, stricter case, since they automatically satisfy absolute summability. Zeros or symmetry of the impulse response don’t determine stability by themselves, so they don’t guarantee BIBO stability the way absolute summability does.

In a discrete-time LTI system, BIBO stability means that every bounded input produces a bounded output. This holds if the impulse response is absolutely summable: the sum over all n of |h[n]| is finite.

Why this works: the output is the convolution y[n] = ∑ h[k] x[n−k]. If the input is bounded by some B, then |y[n]| ≤ B ∑ |h[k]|. For this to be true for every n, the sum ∑ |h[k]| must be finite. That absolute summability is both necessary and sufficient for BIBO stability.

Finite-length impulse responses are a concrete, stricter case, since they automatically satisfy absolute summability. Zeros or symmetry of the impulse response don’t determine stability by themselves, so they don’t guarantee BIBO stability the way absolute summability does.