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BIBO Stability

Module by: Richard Baraniuk

Summary: Explains bounded input, bounded output stability.

BIBO stands for bounded input, bounded output. BIBO stable is a condition such that any bounded input yields a bounded output. This is to say that as long as we input a stable signal, we are guaranteed to have a stable output.

In order to understand this concept, we must first look more closely into exactly what we mean by bounded. A bounded signal is any signal such that there exists a value such that the absolute value of the signal is never greater than some value. Since this value is arbitrary, what we mean is that at no point can the signal tend to infinity.

Figure 1: A bounded signal is a signal for which there exists a constant A A such that ∀t:|ft|<A t f t A
Figure 1 (bibo.png)

Once we have identified what it means for a signal to be bounded, we must turn our attention to the condition a system must posess in order to guarantee that if any bounded signal is passed through the system, a bounded signal will arise on the output. It turns out that a continuous-time LTI system with impulse response ht h t is BIBO stable if and only if

Continuous-Time Condition for BIBO Stability

∫-∞∞|ht|dt<∞ t h t (1)
This is to say that the transfer function is absolutely integrable.

To extend this concept to discrete-time, we make the standard transition from integration to summation and get that the transfer function, hn h n , must be absolutely summable. That is

Discrete-Time Condition for BIBO Stability

∑n=-∞∞|hn|<∞ n h n (2)

Stability and Laplace

Stability is very easy to infer from the pole-zero plot of a transfer function. The only condition necessary to demonstrate stability is to show that the ⅈω ω -axis is in the region of convergence.

Figure 2
Subfigure 2.1: Example of a pole-zero plot for a stable continuous-time system.Subfigure 2.2: Example of a pole-zero plot for an unstable continuous-time system.
Subfigure 2.1 (bibo2.png)Subfigure 2.2 (bibo3.png)

Stability and the Z-Transform

Stability for discrete-time signals in the z-domain is about as easy to demonstrate as it is for continuous-time signals in the Laplace domain. However, instead of the region of convergence needing to contain the ⅈω ω -axis, the ROC must contain the unit circle.

Figure 3
Subfigure 3.1: A stable discrete-time system.Subfigure 3.2: An unstable discrete-time system.
Subfigure 3.1 (bibo4.png)Subfigure 3.2 (bibo5.png)

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