BIBO Stability and Impulse Response
Bounded-input bounded-output stability holds exactly when the impulse response is absolutely integrable, equivalent to left-half-plane poles.
Defining BIBO
A system is bounded-input bounded-output stable if there is a finite bound M such that every input bounded by some constant produces an output bounded by M times that constant. This is the working definition of stability for signals-and-systems analysis.
The impulse-response condition
For an LTI system with impulse response h(t), BIBO stability holds if and only if h(t) is absolutely integrable: the integral of the absolute value of h(t) over all time is finite. Intuitively, if the system's memory of a past impulse dies away fast enough, no bounded input can accumulate into an unbounded output.
Connection to poles
Because the transfer function is the Laplace transform of the impulse response, absolute integrability of h(t) corresponds exactly to all poles having negative real parts. A pole at s = -a with a > 0 gives a term e^(-a*t) whose integral is finite; a pole with positive real part gives a growing exponential whose integral diverges.
Edge cases
- A pole at the origin (pure integrator) is not BIBO stable: a step input, itself bounded, produces a ramp output that grows without bound.
- Imaginary-axis pole pairs give a sinusoidal impulse response that is bounded but not absolutely integrable, so also not BIBO stable.
- These marginal cases are why the stability region is the open left half-plane, boundary excluded.
Discrete-time form
For a discrete system with impulse response h[n], BIBO stability requires that the sum of the absolute values of h[n] be finite, equivalent to all z-plane poles lying strictly inside the unit circle.
BIBO analysis is the rigorous underpinning of the casual statement that stable systems have left-half-plane poles. It also clarifies why integrators, though useful inside feedback loops, are not stable in isolation and must be stabilized by the surrounding loop.