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Encyclopedia > 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. It has been suggested that this article or section be merged with electronics engineering. ... Signal processing is the processing, amplification and interpretation of signals. ... In engineering and mathematics, control theory deals with the behavior of dynamical systems over time. ... The word stability has a number of technical meanings, all related to the common meaning of the word. ... In information theory, a signal is a flow of information. ...

1 Time domain condition
2 Frequency domain condition
- 2.1 Continuous signals
- 2.2 Discrete signals
3 See also

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. In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In the language of mathematics, the impulse response of a linear transformation is the image of Diracs delta function under the transformation. ... In mathematics, the term integrable function refers to a function whose integral may be calculated. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

$int_{-infty}^{infty}{left|mathbf{h}(t)right|dt} = | h |_{1} < infty$

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. The word discrete comes from the Latin word discretus which means separate. ... In the language of mathematics, the impulse response of a linear transformation is the image of Diracs delta function under the transformation. ... In mathematics, the term integrable function refers to a function whose integral may be calculated. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

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

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 $mathbf{y}(n)$ is Discrete mathematics, sometimes called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. ... In electrical engineering, specifically in signal processing and control theory, LTI system theory investigates the effects of a linear, time-invariant system on an arbitrary input signal. ... In the language of mathematics, the impulse response of a linear transformation is the image of Diracs delta function under the transformation. ...

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

where $*$ denotes convolution. Then it follows by the definition of convolution For the computer science usage see convolution (computer science) . In mathematics and in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version...

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

Let $| x |_{infty}$ be the maximum value of $mathbf{x}(n)$ , i.e., the infinity norm. In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

$left|mathbf{y}(n)right| = left|sum_{k=-infty}^{infty}{mathbf{h}(n-k) mathbf{x}(k)}right| le sum_{k=-infty}^{infty}{left|mathbf{h}(n-k)right| left|mathbf{x}(k)right|}$ (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 In mathematics, the triangle inequality states that the distance from A to B to C is never shorter than going directly from A to C. The triangle inequality is a theorem in spaces such as the real numbers, Euclidean space, Lp spaces (p â‰¥ 1) and in all inner product spaces...

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

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. A causal system is a system that depends only on the current and previous inputs. ... In mathematics, a rational function is a ratio of polynomials. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In... In physics, engineering, and mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... A causal system is a system that depends only on the current and previous inputs. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... Abscissa means the x coordinate on an (x, y) graph; the input of a mathematical function against which the output is plotted. ... In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i. ... In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form The most famous of Dirichlet series is which is the Riemann zeta function. ... The S plane is a mathematical domain, where instead of viewing processes in the time domain, modelled with time based functions they are viewed, as equations, in the frequency domain. ...

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

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

$int_{-infty}^{infty}{left|mathbf{h}(t) (e^{sigma + j omega})^{- t} right| dt} = int_{-infty}^{infty}{left|mathbf{h}(t) e^{-s t} right| dt} mbox{ where } s = sigma + j omega mbox{ and Re}(s) = sigma = 0$

The region of convergence must therefore include the imaginary axis. In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

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. A causal system is a system that depends only on the current and previous inputs. ... In mathematics, a rational function is a ratio of polynomials. ... A discrete signal is a signal that has been sampled from a continuous signal. ... In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In... In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ... Illustration of a unit circle. ... A causal system is a system that depends only on the current and previous inputs. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... Illustration of a unit circle. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

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

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

$sum_{n = -infty}^{infty}{left|mathbf{h}(n) (r e^{j omega})^{-n} right|} = sum_{n = -infty}^{infty}{left|mathbf{h}(n) z^{- n} right|} mbox{ where } z = r e^{j omega} mbox{ and } r = |z| = 1$

The region of convergence must therefore include the unit circle. In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In... Illustration of a unit circle. ...

Contents

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