What is the condition for the BIBO stable

BIBO stability (Bounded Input, Bounded Output stability) is a fundamental concept in the analysis of linear time-invariant (LTI) systems, commonly encountered in signal processing and control theory. A system is considered BIBO stable if, for any bounded input signal, the output remains bounded as well.

Condition for BIBO Stability:

For a continuous-time system described by its impulse response h(t)h(t)h(t):

y(t)=∫−∞∞h(τ)⋅x(t−τ) dτy(t) = \int_{-\infty}^{\infty} h(\tau) \cdot x(t - \tau) \, d\tauy(t)=∫−∞∞?h(τ)⋅x(t−τ)dτ

or for a discrete-time system described by its impulse response h[n]h[n]h[n]:

y[n]=∑k=−∞∞h[k]⋅x[n−k]y[n] = \sum_{k=-\infty}^{\infty} h[k] \cdot x[n-k]y[n]=∑k=−∞∞?h[k]⋅x[n−k]

The condition for BIBO stability is that the impulse response h(t)h(t)h(t) (or h[n]h[n]h[n] in discrete-time) must be absolutely integrable (or absolutely summable in discrete-time). Mathematically, this is expressed as:

Continuous-time system:

∫−∞∞?h(τ)? dτ<∞\int_{-\infty}^{\infty} |h(\tau)| \, d\tau < \infty∫−∞∞??h(τ)?dτ<∞

Discrete-time system:

∑k=−∞∞?h[k]?<∞\sum_{k=-\infty}^{\infty} |h[k]| < \infty∑k=−∞∞??h[k]?<∞

Interpretation:

• Bounded Input: If the input signal x(t)x(t)x(t) (or x[n]x[n]x[n] in discrete-time) is bounded, meaning ?x(t)?<M|x(t)| < M?x(t)?<M (or ?x[n]?<M|x[n]| < M?x[n]?<M for discrete-time), where MMM is some finite constant, then the output y(t)y(t)y(t) (or y[n]y[n]y[n]) will also be bounded.

• Impulse Response: The stability of the system hinges on the nature of its impulse response h(t)h(t)h(t) (or h[n]h[n]h[n]). If h(t)h(t)h(t) (or h[n]h[n]h[n]) satisfies the absolute integrability (or summability) condition, then the system will be BIBO stable.

Practical Implications:

• BIBO stability ensures that the system will not amplify input signals beyond a certain limit. This property is essential for ensuring that signal processing systems, filters, and control systems behave predictably and do not generate uncontrollable or unbounded outputs from bounded inputs.

• Engineers and designers use BIBO stability as a criterion when designing systems to guarantee that they will perform reliably under various input conditions without causing instability or unpredictable behavior.

In summary, the condition for BIBO stability is that the impulse response of the system must be integrable (or summable), ensuring that the output remains bounded for any bounded input signal.