Bounded-Input Bounded-Output (BIBO) Stability is an important concept in control systems and signal processing. It describes a system whose output remains bounded whenever the input is bounded.
This type of stability is commonly used to determine whether a linear time-invariant (LTI) system behaves in a stable manner.
Table of Contents
Definition of Bounded-Input Bounded-Output (BIBO) Stability
A system is said to be BIBO stable if every bounded input produces a bounded output.
Mathematically, if the input signal satisfies
\[
|x(t)| \leq M_x
\]
where \(M_x\) is a finite constant, then the output must satisfy
\[
|y(t)| \leq M_y
\]
where \(M_y\) is also finite.
If this condition holds for all bounded inputs, the system is BIBO stable.
Condition for BIBO Stability of LTI Systems
For a linear time-invariant system, the output is given by the convolution of the input with the impulse response.
\[
y(t) = x(t) * h(t)
\]
A continuous-time LTI system is BIBO stable if the impulse response satisfies the condition
\[
\int_{-\infty}^{\infty} |h(t)| \, dt < \infty
\]
This means the absolute value of the impulse response must be integrable.
BIBO Stability for Discrete-Time Systems
For discrete-time systems, the stability condition becomes
\[
\sum_{n=-\infty}^{\infty} |h(n)| < \infty
\]
Thus, the sum of the absolute values of the impulse response must be finite.
BIBO Stability and Transfer Function
For systems described by a transfer function, BIBO stability depends on
the location of the poles.
Continuous-Time Systems
A system is BIBO stable if all poles of the transfer function lie in the left half of the s-plane.
Discrete-Time Systems
A system is BIBO stable if all poles lie inside the unit circle in the z-plane.
Example of a Stable System
Consider the impulse response
\[
h(t) = e^{-at} u(t), \quad a > 0
\]
The integral of the absolute value is
\[
\int_0^{\infty} e^{-at} dt = \frac{1}{a}
\]
Since the value is finite, the system is BIBO stable.
Example of an Unstable System
Consider
\[
h(t) = e^{at} u(t), \quad a > 0
\]
The integral becomes infinite; therefore, the system is not BIBO stable.
Importance of BIBO Stability
BIBO stability is important because it ensures that:
• The system does not produce unbounded outputs
• Signals remain physically realizable
• Communication and control systems operate reliably
Applications
BIBO stability analysis is widely used in:
• Control systems
• Digital signal processing
• Communication systems
• Filter design
• System modeling and analysis
Conclusion
Bounded-Input Bounded-Output stability is a fundamental stability criterion used to evaluate system behavior. A system is BIBO stable if every bounded input results in a bounded output. For LTI systems, this condition is satisfied when the impulse response is absolutely integrable or when the poles of the transfer function lie in the stable region of the complex plane.