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BIBO stability
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- Bibo redirects here. For the Egyptian football player nicknamed Bibo, see Mahmoud El-Khateeb.
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 such that the signal magnitude never exceeds , that is
for discrete-time signals, or
for continuous-time signals.
a class="link1" onMouseover='showByLink("m3169287",this)' onMouseout='hide("m3169287")'href="http://www.absoluteastronomy.com/topics/Continuous_function">continuous time, the condition for BIBO stability is that the impulse response be absolutely integrable, i.e., its L1 norm exist.
a class="link1" onMouseover='showByLink("m3169291",this)' onMouseout='hide("m3169291")'href="http://www.absoluteastronomy.com/topics/Discrete_time">discrete time, the condition for BIBO stability is that the impulse response be absolutely summable, i.e., its norm exist.
n a discrete, linear, time-invariant system with impulse response the relationship between the input and the output is
where denotes convolution.
Then it follows by the definition of convolution
Let be the maximum value of , i.e., the supremum norm.
- (by the triangle inequality)
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If is BIBO stable, then and
So if (i.e., it is bounded) then is bounded as well because .
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.
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Encyclopedia
- Bibo redirects here. For the Egyptian football player nicknamed Bibo, see Mahmoud El-Khateeb.
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 such that the signal magnitude never exceeds , that is
for discrete-time signals, or
for continuous-time signals.
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.
Discrete-time necessary and sufficient condition
In discrete time, the condition for BIBO stability is that the impulse response be absolutely summable, i.e., its norm exist.
Proof of sufficiency
Given a discrete, linear, time-invariant system with impulse response the relationship between the input and the output is
where denotes convolution.
Then it follows by the definition of convolution
Let be the maximum value of , i.e., the supremum norm.
- (by the triangle inequality)
-
-
-
If is BIBO stable, then and
So if (i.e., it is bounded) then is bounded as well because .
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 :
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where and .
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:
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where and .
The region of convergence must therefore include the unit circle.
See also
Further reading
- 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 0133737624
- 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
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