BIBO

BIBO stands for bounded-input, bounded-output. It is a stability notion used in control theory and signal processing. A system is BIBO stable if every bounded input results in a bounded output, for all admissible inputs. It is a property of the system, independent of particular inputs, though it concerns the response to bounded signals.

For continuous-time linear time-invariant systems with impulse response h(t), BIBO stability holds if the impulse response is absolutely integrable: ∫_{-∞}^{∞} |h(t)| dt < ∞. For discrete-time LTI systems with impulse response h[n], the condition is ∑_{n=-∞}^{∞} |h[n]| < ∞. In many practical cases, causality reduces the ranges to t ≥ 0 or n ≥ 0.

In the transfer-function framework, continuous-time H(s) must have all poles in the open left half-plane (no poles on the imaginary axis) for BIBO stability; discrete-time H(z) must have all poles strictly inside the unit circle. When these pole conditions hold, the impulse response is absolutely integrable or summable, ensuring bounded outputs for bounded inputs.

Common examples: h(t) = e^{-a t} u(t) with a > 0 is BIBO stable. In discrete time, h[n] = a^n u[n] with |a| < 1 is BIBO stable. Systems with poles on the imaginary axis or outside the stability region are not BIBO stable; they may exhibit growing or unbounded responses to bounded inputs.