BIBO stability

In electrical engineering, specifically signal processing and control theory, BIBO Stability is a form of stability for signals and systems. 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 the signal is finite valued for all times ( $h$ ^* < \infty \quad \forall n \in \mathbb{Z} or $h(t) < \infty \quad \forall t \in \mathbb{R}$ ).

Time domain condition

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|\mathbf{h}(t)\right|dt} = \| h \|_{1} < \infty

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

\ell^1

norm exist.

\sum_{n=-\infty}^{\infty}{\left|\mathbf{h}(n)\right|} = \| h \|_{1} < \infty

Proof of sufficiency

Given a discrete, linear, time-invariant system with impulse response

\mathbf{h}(n)

the relationship between the input

\mathbf{x}(n)

and the output

\mathbf{y}(n)

\mathbf{y}(n) = \mathbf{h}(n) * \mathbf{x}(n)

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

\mathbf{y}(n) = \sum_{k=-\infty}^{\infty}{\mathbf{h}(k) \mathbf{x}(n-k)}

Let $\| x \|_{\infty}$ be the maximum value of $\mathbf{x}(n)$ , i.e., the infinity norm.

\left|\mathbf{y}(n)\right| = \left|\sum_{k=-\infty}^{\infty}{\mathbf{h}(n-k) \mathbf{x}(k)}\right|

\le \sum_{k=-\infty}^{\infty}{\left|\mathbf{h}(n-k)\right| \left|\mathbf{x}(k)\right|}

(by the triangle inequality)

\le \sum_{k=-\infty}^{\infty}{\left|\mathbf{h}(n-k)\right| \| x \|_{\infty}}

= \| x \|_{\infty} \sum_{k=-\infty}^{\infty}{\left|\mathbf{h}(n-k)\right|}

= \| x \|_{\infty} \sum_{k=-\infty}^{\infty}{\left|\mathbf{h}(k)\right|}

If $\mathbf{h}(n)$ is BIBO stable, then $\sum_{k=-\infty}^{\infty}{\left|\mathbf{h}(k)\right|} = \| h \|_1 < \infty$ and

\| x \|_{\infty} \sum_{k=-\infty}^{\infty}{\left|\mathbf{h}(k)\right|} = \| x \|_{\infty} \| h \|_1

So if $\| x \|_{\infty} < \infty$ (i.e., it is bounded) then $\left|\mathbf{y}(n)\right|$ is bounded as well because $\| x \|_{\infty} \| h \|_1 < \infty$ .

The proof for continuous-time follows the same arguments.

Frequency domain condition

Continuous 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|\mathbf{h}(t)\right| dt}

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

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

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

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

where $s = \sigma + j \omega$ and $\mbox{Re}(s) = \sigma = 0$ .

The region of convergence must therefore include the imaginary axis.

Discrete 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 derivation:

\sum_{n = -\infty}^{\infty}{\left|\mathbf{h}(n)\right|}

= \sum_{n = -\infty}^{\infty}{\left|\mathbf{h}(n)\right| \left| e^{-j \omega n} \right|}

= \sum_{n = -\infty}^{\infty}{\left|\mathbf{h}(n) (1 \cdot e)^{-j \omega n} \right|}

=\sum_{n = -\infty}^{\infty}{\left|\mathbf{h}(n) (r e^{j \omega})^{-n} \right|}

= \sum_{n = -\infty}^{\infty}{\left|\mathbf{h}(n) z^{- n} \right|}

where $z = r e^{j \omega}$ and $r = |z| = 1$ .

The region of convergence must therefore include the unit circle.

References

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. Mandalokis Digital Signal Processing Principals, Algorithms and Applications third edition, Prentice Hall, 1996, ISBN 0-13-394338-9
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

Control theory | Signal processing | Digital signal processing

BIBO-Stabilität | Stabilité EBSB

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