Name: lti system tricks for easier convolution
Uploaded: 2024-09-26T13:16:55.000Z
Duration: 1 min 20 s
Channel: JoVE

利用卷积计算的固有性质可以简化这些计算。
交换律表明，线性时不变系统中的输入和冲激响应可以互换，而不会影响输出。
结合律表明，三个函数的合并卷积在卷积执行顺序不同的情况下仍保持不变。
当两个具有冲激响应的线性时不变系统串联连接时，可利用结合律将它们各自的方程合并，从而推导出一个等效的联合冲激响应，该响应类似于各自冲激响应的卷积。
分配律使得多个输入信号之和的卷积运算成为可能，并通过将复杂的脉冲响应分解为更简单的组成部分来简化分析。
时移性质表明，延迟一个时不变系统的输入，或当系统本身具有内建延迟时，会导致输出延迟两个延迟之和。
从计算角度来看，该特性允许对信号进行延迟或提前，以利用其对称性或因果性，从而简化卷积运算。

卷积属性-i

卷积计算可以利用其固有属性来简化。

交换性表明 LTI（线性时不变）系统的输入和脉冲响应可以互换而不会影响输出：

<img alt="Equation1" src="/files/ftp_upload/16050/16050_PT%20EQUATION_1.svg" style="height: 16px;">

结合性表明，无论卷积顺序如何，三个函数的合并卷积都保持不变。例如，对于三个函数 x(t)、h_1(t) 和 h_2(t)，写为：

<img alt="Equation2" src="/files/ftp_upload/16050/16050_PT%20EQUATION_2.svg" style="height: 16px;">

当两个具有脉冲响应的 LTI 系统串联时，可以使用结合性将它们各自的方程组合起来，以得出等效的联合脉冲响应，即它们各自脉冲响应的卷积。

分配性特性使卷积运算能够对多个输入信号的总和进行，从而将复杂的脉冲响应分解为更简单的组件。从数学上讲，这表示为：

<img alt="Equation3" src="/files/ftp_upload/16050/16050_PT%20EQUATION_3.svg" style="height: 16px;">

时间移位特性意味着延迟时不变系统的输入会导致输出延迟相同的量。同样，如果系统具有内置延迟，则输出将延迟输入延迟和系统延迟的总和。对于时间移位 t_0：

<img alt="Equation4" src="/files/ftp_upload/16050/16050_PT%20EQUATION_4.svg" style="height: 16px;">

从计算上讲，此属性允许信号延迟或提前，利用其对称性或因果关系来简化卷积运算。

这些属性（交换性、结合性、分配性和时间移位）是简化 LTI 系统中卷积运算的基本工具，使复杂的信号处理任务更易于管理和更高效。

卷积属性-I

卷积计算可以通过利用其固有特性来简化。  
交换律表明，输入信号与冲激响应……

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 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 you combine their equations into one equivalent impulse response.

The equivalent joint impulse response is found by convolving the two individual impulse responses. The distributive property also helps by allowing convolution to be applied to a sum of input signals. This makes it easier to break a complex impulse response into simpler parts.

The time-shift property shows what happens when a signal is delayed. If the input of a time-invariant system is delayed, the output is delayed by the same amount. If the system already has a delay, the total output delay is the input delay plus the system delay. This same property can also help when signals are shifted forward or backward, especially when symmetry or causality makes the convolution simpler.

These properties include commutative, associative, distributive, and time-shift rules. Together, they are basic tools for simplifying convolution in LTI systems. They make signal processing tasks more manageable and efficient.

LTI System Tricks for Easier 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 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

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...

key terms and concepts

Learn how commutative, associative, distributive, and time-shift properties simplify convolution in LTI systems for high school signal processing students.

LTI System Tricks for Easier 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

LTI System Tricks for Easier Convolution

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