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

可以利用卷积来确定系统的稳定性。
如果一个输入信号具有一个常数，使得该信号的模在任何时刻都不会超过该常数，则该信号被视为有界的。
一个系统如果对任意有界输入信号始终产生有界输出，则称该系统为有界输入有界输出稳定，也称为 BIBO 稳定。图中展示了 BIBO 稳定与非 BIBO 稳定系统的示例。
当有界连续时间输入信号施加于线性时不变系统时，可利用卷积积分来评估该系统的有界输入-有界输出（BIBO）稳定性。
输入信号的有界性可通过一个常数表示，而卷积积分公式则用于确定有界输出。
如果被积函数是有限的，则输出也是有限的，这意味着具有绝对可积冲激响应的连续时间系统是BIBO稳定的。
这一相同的过程也适用于离散时间系统；通过卷积积分可获得有界输出。
如果求和项具有有限值，则输出是有限的，这表明离散时间系统的冲激响应若绝对可和，则该系统是BIBO稳定的。

连续和离散时间系统的 bibo 稳定性

BIBO stability of continuous and discrete -time systems

系统稳定性是信号处理中的一个基本概念，通常使用卷积进行评估。对于被视为有界输入有界输出 (BIBO) 稳定的系统，任何有界输入信号都必须产生有界输出信号。有界输入信号是指模数在任何时间点不超过某个常数的信号。

为了确定 BIBO 稳定性，当有界连续时间输入应用于线性时不变 (LTI) 系统时，使用卷积积分。输入信号的有界性由常数表示，卷积积分有助于确定输出是否保持有界。从数学上讲，这意味着如果卷积积分的被积函数是有限的，则输出也将是有限的。具体而言，如果连续时间系统的脉冲响应是可积的，则该系统是 BIBO 稳定的，这意味着脉冲响应的绝对值的积分是有限的，表示为

<img alt="Equation1" src="/files/ftp_upload/16053/16053_PT%20EQUATION_1.svg" style="height: 36px;">

此条件确保输出对于任何有界输入信号都保持在界限内，从而确认系统的稳定性。

同样的原则也适用于离散时间系统。离散时间系统中的 BIBO 稳定性是通过卷积级数的求和来确定的。对于离散时间系统，如果求和项具有有限值，则输出是有限的，这表明如果系统的脉冲响应是可求和的，则该系统是 BIBO 稳定的，如下面的表达式所示。

<img alt="Equation2" src="/files/ftp_upload/16053/16053_PT%20EQUATION_2.svg" style="height: 28px;">

换句话说，如果脉冲响应的绝对值之和是有限的，则系统将对任何有界输入产生有界输出，从而确认 BIBO 稳定性。

BIBO 稳定性的重要性在于其在实际系统中的应用，其中确保输出在有界输入的情况下保持在可接受的范围内至关重要。理解和应用卷积和脉冲响应积分的概念对于设计和分析连续和离散时间域中的稳定系统至关重要。

连续和离散时间系统的 BIBO 稳定性

系统的稳定性可通过卷积来确定。  
若输入信号包含一个常数，可确保该信号的模值始终不超过……

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

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Linear Time- Invariant Systems

线性时不变系统

KEY TERMS AND CONCEPTS

关键术语与概念

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

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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...

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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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