BIBO stability: Information from Answers.com

Bibo redirects here. For the Egyptian football player nicknamed Bibo, see Mahmoud El-Khateeb.

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In electrical engineering, specifically signal processing and control theory, BIBO stability is a form of stability for linear signals and systems that take inputs. BIBO stands for Bounded-Input Bounded-Output. If a system is BIBO stable, then the output will be bounded for every input to the system that is bounded.

A signal is bounded if there is a finite value $B > 0$ such that the signal magnitude never exceeds $B$ , that is

$\ |h[n]| \leq B \quad \forall n \in \mathbb{Z}$ for discrete-time signals, or

$\ |h(t)| \leq B \quad \forall t \in \mathbb{R}$ for continuous-time signals.

1 Time-domain condition for linear time invariant systems
2 Frequency-domain condition for linear time invariant systems
- 2.1 Continuous-time signals
- 2.2 Discrete-time signals
3 See also
4 Further reading
5 References

Time-domain condition for linear time invariant systems

Continuous-time necessary and sufficient condition

In continuous time, the condition for BIBO stability is that the impulse response be absolutely integrable, i.e., its L¹ norm exist.

$\int_{-\infty}^{\infty}{\left|h(t)\right|\,\operatorname{d}t} = \| h \|_{1} < \infty$

Discrete-time sufficient condition

In discrete time, the condition for BIBO stability is that the impulse response be absolutely summable, i.e., its $\ell^1$ norm exist.

$\ \sum_{n=-\infty}^{\infty}{\left|h[n]\right|} = \| h \|_{1} < \infty$

Proof of sufficiency

Given a discrete, linear, time-invariant system with impulse response $\ h[n]$ the relationship between the input $\ x[n]$ and the output $\ y[n]$ is

$\ y[n] = h[n] * x[n]$

where $*$ denotes convolution. Then it follows by the definition of convolution

$\ y[n] = \sum_{k=-\infty}^{\infty}{h[k] x[n-k]}$

Let $\| x \|_{\infty}$ be the maximum value of $\ |x[n]|$ , i.e., the supremum norm.

$\left|y[n]\right| = \left|\sum_{k=-\infty}^{\infty}{h[n-k] x[k]}\right|$

$\le \sum_{k=-\infty}^{\infty}{\left|h[n-k]\right| \left|x[k]\right|}$ (by the triangle inequality)

$\le \sum_{k=-\infty}^{\infty}{\left|h[n-k]\right| \| x \|_{\infty}}$

$= \| x \|_{\infty} \sum_{k=-\infty}^{\infty}{\left|h[n-k]\right|}$

$= \| x \|_{\infty} \sum_{k=-\infty}^{\infty}{\left|h[k]\right|}$

If $h [n]$ is absolutely summable, then $\sum_{k=-\infty}^{\infty}{\left|h[k]\right|} = \| h \|_1 < \infty$ and

$\| x \|_{\infty} \sum_{k=-\infty}^{\infty}{\left|h[k]\right|} = \| x \|_{\infty} \| h \|_1$

So if $h [n]$ is absolutely summable and $\left|x[n]\right|$ is bounded, then $\left|y[n]\right|$ is bounded as well because $\| x \|_{\infty} \| h \|_1 < \infty$ .

The proof for continuous-time follows the same arguments.

Frequency-domain condition for linear time invariant systems

Continuous-time signals

For a causal, rational, continuous-time system, the condition for stability is that the region of convergence (ROC) of the Laplace transform includes the imaginary axis. When the system is causal, the ROC is the open region to the right of a vertical line whose abscissa is the real part of the largest pole. (Largest here is defined so that the real part of the largest pole is greater than the real part of any other pole in the system.) The real part of the largest pole defining the ROC is called the abscissa of convergence. Therefore, all poles of the system must be in the strict left half of the s-plane for BIBO stability.

This stability condition can be derived from the above time-domain condition as follows :

$\int_{-\infty}^{\infty}{\left|h(t)\right| \,\operatorname{d}t}$

$= \int_{-\infty}^{\infty}{\left|h(t)\right| \left| e^{-j \omega t} \right| dt}$

$= \int_{-\infty}^{\infty}{\left|h(t) (1 \cdot e)^{-j \omega t} \right| dt}$

$= \int_{-\infty}^{\infty}{\left|h(t) (e^{\sigma + j \omega})^{- t} \right| dt}$

$= \int_{-\infty}^{\infty}{\left|h(t) e^{-s t} \right| dt}$

where $s = σ + j ω$ and $Re(s) = σ = 0$ .

The region of convergence must therefore include the imaginary axis.

Discrete-time signals

For a causal, rational, discrete time system, the condition for stability is that the region of convergence (ROC) of the z-transform includes the unit circle. When the system is causal, the ROC is the open region outside a circle whose radius is the magnitude of the pole with largest magnitude. Therefore, all poles of the system must be inside the unit circle in the z-plane for BIBO stability.

This stability condition can be derived in a similar fashion to the continuous-time derivation:

$\sum_{n = -\infty}^{\infty}{\left|h[n]\right|} = \sum_{n = -\infty}^{\infty}{\left|h[n]\right| \left| e^{-j \omega n} \right|}$