Name: convolution rules for width and area
Uploaded: 2024-09-26T13:16:55.000Z
Duration: 1 min 17 s
Channel: JoVE

重要的卷积性质包括宽度、面积、微分和积分性质。
宽度属性表明，如果输入信号的持续时间分别为 T1 和 T2，则输出响应的宽度等于这两个持续时间之和，而与两个函数的形状无关。例如，将两个持续时间分别为 2s 和 1s 的矩形脉冲进行卷积，会得到一个宽度为 3s 的函数。
面积性质表明，两个函数卷积下的面积等于各自函数下面积的乘积。
微分性质表明，卷积的导数等于输入信号的导数与冲激响应的卷积，或者反过来，输入信号与冲激响应的导数的卷积。
结合这些方程并推广至高阶导数，可得到通用的微分关系。
对于给定的卷积，其积分性质可表示为冲激响应与输入响应的卷积。这一性质也可按相反顺序表述。
若使用阶跃函数进行卷积，则该线性时不变系统表现得如同理想积分器。

卷积属性-ii

重要的卷积属性包括宽度、面积、微分和积分属性。

宽度属性表示，如果输入信号的持续时间为 T_1 和 T_2，则输出响应的宽度等于两个持续时间的总和，而与两个函数的形状无关。例如，卷积两个持续时间分别为 2 秒和 1 秒的矩形脉冲将产生一个宽度为 3 秒的函数。

面积属性断言两个函数卷积下的面积等于每个函数下面积的乘积。从数学上讲，如果 x(t) 和 h(t) 是两个函数，则面积属性的表达式可以表示为：

<img alt="Equation1" src="/files/ftp_upload/16051/16051_PT%20EQUATION_1.svg" style="height: 16px;">

微分属性表示卷积的导数等于输入信号的导数和脉冲响应的卷积，反之亦然。这可以表示为：

<img alt="Equation2" src="/files/ftp_upload/16051/16051_PT%20EQUATION_2.svg" style="height: 34px;">

将这些方程组合起来并对高阶导数进行推广，可得到一般微分关系。

积分性质表明卷积的积分产生脉冲响应和输入信号积分的卷积，反之亦然。这在数学上表示为：

<img alt="Equation3" src="/files/ftp_upload/16051/16051_PT%20EQUATION_3.svg" style="height: 42px;">

如果使用阶跃函数执行卷积，则 LTI 系统的行为就像一个理想的积分器。

这些性质（宽度、面积、微分和积分）对于简化和理解 LTI 系统中的卷积运算至关重要，从而可以更轻松地分析和设计复杂的信号处理任务。

卷积属性-II

重要的卷积性质包括宽度性质、面积性质、微分性质和积分性质。  
宽度性质表明，若……的持续时间分别为……

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

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 seconds and 1 second gives an output with a width of 3 seconds.

The area property states that the area under the convolution of two functions equals the product of their individual areas. In other words, if x(t) and h(t) are convolved, the total area of the result depends on the area of each input. The differentiation property says that the derivative of a convolution is the convolution of the derivative of the input signal and the impulse response, or the other way around.

The integration property gives the matching rule for integrals. It says that integrating a convolution produces the convolution of the impulse response and the integral of the input signal, or vice versa. If a convolution is performed with a step function, the LTI system acts like an ideal integrator. Together, these width, area, differentiation, and integration rules help simplify convolution in LTI systems and support the analysis and design of more complex signal processing tasks.

Convolution Rules for Width and Area

linear time-invariant systems

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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 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

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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...

key terms and concepts

Learn the key convolution rules for width, area, differentiation, and integration in LTI systems, with a simple rectangular pulse example and step-function behavior.

Convolution Rules for Width and Area | 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

Convolution Rules for Width and Area

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