Editing BIBO stability

{{short description|When a system's outputs are bounded for every bounded input}}
{{No footnotes|date=April 2009}}

In [[signal processing]], specifically [[control theory]], '''bounded-input, bounded-output''' ('''BIBO''') '''stability''' is a form of [[Control theory#Stability|stability]] for [[Signal (information theory)|signal]]s and [[Control system|systems]] that take inputs. If a system is BIBO stable, then the output will be [[Bounded function|bounded]] for every input to the system that is bounded.

A signal is bounded if there is a finite value <math>B > 0</math> such that the signal magnitude never exceeds <math>B</math>, that is
:For [[discrete-time]] signals: <math>\exists B \forall n(\ |y[n]| \leq B) \quad n \in \mathbb{Z}</math>
:For [[continuous-time]] signals: <math>\exists B \forall t(\ |y(t)| \leq B) \quad t \in \mathbb{R}</math>

== Time-domain condition for linear time-invariant systems==
===Continuous-time necessary and sufficient condition===
For a [[continuous function|continuous time]] [[LTI system theory|linear time-invariant (LTI)]] system, the condition for BIBO stability is that the [[impulse response]], <math> h(t)</math> , be [[P-integrable function|absolutely integrable]], i.e., its [[Lp space|L<sup>1</sup> norm]] exists.

: <math> \int_{-\infty}^\infty \left|h(t)\right|\,\mathord{\operatorname{d}}t = \| h \|_1 \in \mathbb{R}</math>

===Discrete-time sufficient condition===
For a [[discrete time]] LTI system, the condition for BIBO stability is that the [[impulse response]] be [[P-integrable function|absolutely summable]], i.e., its <math>\ell^1</math> [[Lp space|norm]] exists.

:<math>\ \sum_{n=-\infty}^\infty |h[n]| = \| h \|_1 \in \mathbb{R}</math>

====Proof of sufficiency====
Given a [[discrete mathematics|discrete]] time LTI system with [[impulse response]] <math>\ h[n]</math> the relationship between the input <math>\ x[n]</math> and the output <math>\ y[n]</math> is

:<math>\ y[n] = h[n] * x[n]</math>

where <math>*</math> denotes [[convolution]]. Then it follows by the definition of convolution

:<math>\ y[n] = \sum_{k=-\infty}^\infty h[k] x[n-k]</math>

Let <math>\| x \|_{\infty}</math> be the maximum value of <math>\ |x[n]|</math>, i.e., the [[Supremum norm|<math>L_{\infty}</math>-norm]].

:<math>\left|y[n]\right| = \left|\sum_{k=-\infty}^\infty h[n-k] x[k]\right|</math>

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

: <math>
\begin{align}
& \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|
\end{align}
</math>

If <math>h[n]</math> is absolutely summable, then <math>\sum_{k=-\infty}^{\infty}{\left|h[k]\right|} = \| h \|_1  \in \mathbb{R}</math> and

:<math>\| x \|_\infty \sum_{k=-\infty}^\infty \left|h[k]\right| = \| x \|_\infty \| h \|_1</math>

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

The proof for continuous-time follows the same arguments.

== Frequency-domain condition for linear time-invariant systems==
=== Continuous-time signals ===

For a [[rational function|rational]] and [[continuous function|continuous-time system]], the condition for stability is that the [[region of convergence]] (ROC) of the [[Laplace transform]] includes the [[complex plane|imaginary axis]]. When the system is [[Causal system|causal]], the ROC is the [[open region]] to the right of a vertical line whose [[abscissa]] is the [[real part]] of the "largest pole", or the [[pole (complex analysis)|pole]] that has the greatest real part of any 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:

:<math>
\begin{align}
\int_{-\infty}^\infty \left|h(t)\right| \, dt
& = \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
\end{align}
</math>

where <math>s = \sigma + j \omega</math> and <math>\operatorname{Re}(s) = \sigma = 0.</math>

The [[region of convergence]] must therefore include the [[complex plane|imaginary axis]].

=== Discrete-time signals ===

For a [[rational function|rational]] and [[discrete signal|discrete time system]], the condition for stability is that the [[Laplace transform#Region of convergence|region of convergence]] (ROC) of the [[z-transform]] includes the [[unit circle]]. When the system is [[Causal system|causal]], the ROC is the [[open region]] outside a circle whose radius is the magnitude of the [[pole (complex analysis)|pole]] with largest magnitude. Therefore, all poles of the system must be inside the [[unit circle]] in the [[Z-transform|z-plane]] for BIBO stability.

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

:<math>
\begin{align}
\sum_{n = -\infty}^\infty \left|h[n]\right|
& = \sum_{n = -\infty}^\infty \left|h[n]\right| \left| e^{-j \omega n} \right| \\
& = \sum_{n = -\infty}^\infty \left|h[n] (1 \cdot e)^{-j \omega n} \right| \\
& =\sum_{n = -\infty}^\infty \left|h[n] (r e^{j \omega})^{-n} \right| \\
& = \sum_{n = -\infty}^\infty \left|h[n] z^{- n} \right|
\end{align}
</math>

where <math>z = r e^{j \omega}</math> and <math>r = |z| = 1</math>.

The [[Laplace transform#Region of convergence|region of convergence]] must therefore include the [[unit circle]].

== See also ==
* [[LTI system theory]]
* [[Finite impulse response|Finite impulse response (FIR) filter]]
* [[Infinite impulse response|Infinite impulse response (IIR) filter]]
* [[Nyquist plot]]
* [[Routh–Hurwitz stability criterion]]
* [[Bode plot#Gain margin and phase margin|Bode plot]]
* [[Phase margin]]
* [[Root locus|Root locus method]]
* [[Input-to-state stability]]

==Further reading==
{{refbegin}}
*Gordon E. Carlson ''Signal and Linear Systems Analysis with Matlab'' second edition, Wiley, 1998, {{ISBN|0-471-12465-6}}
*John G. Proakis and Dimitris G. Manolakis ''Digital Signal Processing Principals, Algorithms and Applications'' third edition, Prentice Hall, 1996, {{ISBN|0-13-373762-4}}
*D. Ronald Fannin, William H. Tranter, and Rodger E. Ziemer ''Signals & Systems Continuous and Discrete'' fourth edition, Prentice Hall, 1998, {{ISBN|0-13-496456-X}}
*[http://cnx.org/content/m12319/latest/ Proof of the necessary conditions for BIBO stability.]
*Christophe Basso ''Designing Control Loops for Linear and Switching Power Supplies: A Tutorial Guide'' first edition, Artech House, 2012, 978-1608075577
*{{cite journal|author=Michael Unser|journal=IEEE Transactions on Signal Processing|volume=68|date=2020|pages=5904–5913|title=A Note on BIBO Stability|doi=10.1109/TSP.2020.3025029|arxiv=2005.14428|bibcode=2020ITSP...68.5904U }}
{{refend}}

==References==
{{reflist}}

[[Category:Signal processing]]
[[Category:Digital signal processing]]
[[Category:Articles containing proofs]]
[[Category:Stability theory]]