Name: testing system stability with convolution
Uploaded: 2024-09-26T13:16:56.000Z
Duration: 1 min 24 s
Channel: JoVE

La estabilidad de los sistemas se puede determinar mediante convolución.
Si una señal de entrada tiene una constante que garantiza que el módulo de la señal nunca la exceda en ningún momento en el tiempo, se considera acotada.
Un sistema es de entrada acotada y salida acotada, también conocida como BIBO estable, si cualquier señal de entrada acotada produce invariablemente una salida acotada. La figura ilustra ejemplos de sistemas BIBO-estables y no BIBO-estables.
La integral de convolución se puede utilizar para evaluar la estabilidad de BIBO cuando se aplica una entrada de tiempo continuo acotada a un sistema lineal invariante en el tiempo.
La acotación de la señal de entrada se puede representar a través de una constante, y las formulaciones integrales de convolución determinan la salida acotada.
Si el integrando es finito, la salida es finita, lo que significa que un sistema de tiempo continuo con una respuesta de impulso absolutamente integrable es BIBO estable.
Este mismo proceso se aplica a los sistemas de tiempo discreto; una salida acotada se logra a través de la integral de convolución.
La salida es finita si el término de suma tiene un valor finito, lo que indica que un sistema de tiempo discreto es BIBO estable si su respuesta al impulso es absolutamente sumable.

estabilidad bibo de sistemas de tiempo continuo y discreto

La estabilidad del sistema es un concepto fundamental en el procesamiento de señales, que a menudo se evalúa mediante convolución. Para que un sistema se considere estable con entrada limitada y salida limitada (BIBO), cualquier señal de entrada limitada debe producir una señal de salida limitada. Una señal de entrada limitada es aquella en la que el módulo no excede una cierta constante en ningún momento.
Para determinar la estabilidad (BIBO), se utiliza la integral de convolución cuando se aplica una entrada de tiempo continuo limitada a un sistema lineal invariante en el tiempo (LTI). La acotación de la señal de entrada se representa mediante una constante, y la integral de convolución ayuda a determinar si la salida permanece limitada. Matemáticamente, esto significa que si el integrando de la integral de convolución es finito, la salida también será finita. En concreto, un sistema de tiempo continuo es estable (BIBO) si su respuesta al impulso es integrable, lo que significa que la integral del valor absoluto de la respuesta al impulso es finita y se representa como,
<img alt="Equation1" src="/files/ftp_upload/16053/16053_PT%20EQUATION_1.svg" style="text-align: center; height: 36px;">
Esta condición garantiza que la salida se mantenga dentro de los límites para cualquier señal de entrada acotada, lo que confirma la estabilidad del sistema.
El mismo principio se aplica a los sistemas de tiempo discreto. La estabilidad (BIBO) en sistemas de tiempo discreto se determina mediante la suma de las series de convolución. Para un sistema de tiempo discreto, la salida es finita si el término de suma tiene un valor finito, lo que indica que el sistema es estable (BIBO) si su respuesta al impulso es sumable como se indica en la expresión siguiente.
<img alt="Equation2" src="/files/ftp_upload/16053/16053_PT%20EQUATION_2.svg" style="text-align: center; height: 28px;">
En otras palabras, si la suma de los valores absolutos de la respuesta al impulso es finita, el sistema producirá una salida acotada para cualquier entrada acotada, lo que confirma la estabilidad (BIBO).
La importancia de la estabilidad de (BIBO) radica en su aplicación a sistemas del mundo real, donde es crucial garantizar que las salidas se mantengan dentro de límites aceptables en respuesta a entradas limitadas. Comprender y aplicar los conceptos de convolución e integrales Dde respuesta al impulso son vitales para diseñar y analizar sistemas estables tanto en dominios de tiempo continuo como discreto.

Estabilidad BIBO de sistemas de tiempo continuo y discreto

La estabilidad de los sistemas se puede determinar mediante convolución.
Si una señal de entrada tiene una constante que asegura que el módulo de la señal nunca exceda ...

bibo-stability-of-continuous-and-discrete-time-systems

Education

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JoVE Core

Electrical Engineering

Ingeniería Eléctrica

Unit 1

Unidad 1

Linear Time- Invariant Systems

Sistemas Lineales e Invariantes en el Tiempo

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

BIBO stability of continuous and discrete -time systems

testing system stability with convolution

System stability in signal processing is often tested with convolution. A system is BIBO stable, or bounded-input bounded-output stable, when every bounded input signal produces a bounded output signal. A bounded input signal has a modulus that never exceeds a set constant at any time.

For a continuous-time Linear Time-Invariant, or LTI, system, BIBO stability is checked with the convolution integral. The input is bounded by a constant, and the convolution integral shows whether the output stays bounded too. If the integrand in the convolution integral remains finite, the output remains finite as well. A continuous-time system is BIBO stable when its impulse response is integrable, which means the integral of the absolute value of the impulse response is finite.

The same idea applies to discrete-time systems. Here, stability is checked through the convolution series rather than the convolution integral. A discrete-time system is BIBO stable if the summation is finite, which means its impulse response is summable. In other words, if the sum of the absolute values of the impulse response is finite, any bounded input will still produce a bounded output.

BIBO stability matters in real-world system design and analysis. It helps ensure that outputs stay within acceptable limits when inputs are bounded. Understanding convolution and impulse response conditions is key for studying stable continuous-time and discrete-time systems.

Testing System Stability with Convolution

testing-system-stability-with-convolution

linear time-invariant systems

Impulse Response in LTI Systems

Linear time-invariant systems are defined by linear elements and constant parameters. A system is linear when it shows homogeneity and additivity. These two ideas together are called the superposition property. Superposition is a key rule for analyzing linear systems. For a linear time-invariant system, the response to an impulse at the input can describe the whole input-output behavior. An impulse is a very short, sudden input. Once the impulse response is known, the response to other inputs...

impulse response

RC Circuit Impulse Response

RC circuit impulse response describes how the capacitor voltage changes after a brief input signal. In this system, the voltage source is the input, and the capacitor voltage is the output. The state of the circuit before and after excitation is defined separately so the response can be tracked clearly. Kirchhoff’s law is used to write the input signal equation for the RC circuit. The capacitor current and voltage provide the output relation. After substituting the current expression and...

convolution: math, graphics, and discrete signals

convolution-math-graphics-and-discrete-signals

Convolution in LTI Systems and Signal Response

Convolution describes how a linear time-invariant, or LTI, system responds to an input signal. In an LTI system, the convolution of two signals is written with a convolution operator. The method assumes that all initial conditions are zero. The convolution integral can be split into two parts. These are the zero-input response, also called the natural response, and the zero-state response, also called the forced response. The initial time is marked as t0. To make the integral easier to work...

convolution properties i

LTI System Tricks for Easier Convolution

LTI system properties can make convolution easier to work with. LTI stands for linear time-invariant. In these systems, the input signal and the impulse response can be handled in helpful ways that simplify the final output. The commutative property means the input and the impulse response can be switched without changing the result. The associative property says that when three functions are convolved, the order of grouping does not change the answer. For two LTI systems in series, this lets...

convolution properties ii

Convolution Rules for Width and Area

Convolution rules help describe how an output signal changes when two signals are combined in an LTI system. The main rules here are width, area, differentiation, and integration. These rules make convolution easier to analyze and use in signal processing. The width property says that if two input signals last for T1 and T2, the output response lasts for T1 plus T2. This is true no matter what shape the two functions have. For example, convolving two rectangular pulses with durations of 2...

deconvolution

Finding an Impulse Response from Signals

Deconvolution is a method for finding a system’s impulse response from known input and output signals. It is also called inverse filtering. This technique is useful when the system’s characteristics are not known directly and must be inferred from the signals that can be observed. One common way to do deconvolution is polynomial division. In this approach, the input and output sequences are treated as coefficients of polynomials written in descending order. Long division is then used to divide...

bibo stability of continuous and discrete -time systems

System stability in signal processing is often tested with convolution. A system is BIBO stable, or bounded-input bounded-output stable, when every bounded input signal produces a bounded output signal. A bounded input signal has a modulus that never exceeds a set constant at any time. For a continuous-time Linear Time-Invariant, or LTI, system, BIBO stability is checked with the convolution integral. The input is bounded by a constant, and the convolution integral shows whether the output...

key terms and concepts

Learn how convolution is used to test BIBO stability in continuous-time and discrete-time systems by checking impulse response integrals and sums.

Testing System Stability with Convolution | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Testing System Stability with Convolution

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