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

系统的稳定性可以使用卷积来确定。
如果 input 信号具有一个常数，可确保信号的模数在任何时间点都不会超过它，则认为它是有界的。
系统是有界输入有界输出，如果任何给定的有界输入信号总是产生有界输出，则也称为 BIBO 稳定。该图说明了 BIBO 稳定和非 BIBO 稳定系统的示例。
当有界连续时间输入应用于线性时不变系统时，卷积积分可用于评估 BIBO 稳定性。
输入信号的有界性可以通过常量表示，卷积积分公式确定有界输出。
如果被积函数是有限的，则输出是有限的，这意味着具有绝对可积脉冲响应的连续时间系统是 BIBO 稳定的。
同样的过程也适用于离散时间系统;有界输出是通过卷积积分实现的。
如果求和项具有有限值，则输出是有限的，这表明如果离散时间系统的脉冲响应是绝对可求和的，则它是 BIBO 稳定的。

连续和离散时间系统的 bibo 稳定性

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

系统的稳定性可以使用卷积来确定。
如果输入信号具有确保信号模数永远不会超过 ...

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

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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