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

중요한 컨볼루션 속성은 width, area, differentiation 및 integration 속성입니다.
width 속성은 입력 신호의 duration시간이 각각 T1과 T2인 경우, 출력 응답의 너비가 두 함수의 형태에 관계없이 두 duration의 합과 같음을 나타냅니다. 예를 들어, 지속 시간이 2s와 1s인 두 개의 직사각형 펄스를 컨볼루션하면 너비가 3s인 함수가 생성됩니다.
area 속성은 두 함수의 컨볼루션 아래에 있는 영역이 각 함수에 있는 영역의 곱과 같다고 주장합니다.
differentiation 속성은 컨볼루션의 도함수가 입력 신호의 미분 및 임펄스 응답의 컨볼루션과 같거나 그 반대의 경우도 마찬가지임을 나타냅니다.
방정식을 결합하고 고차 도함수에 대해 일반화하면 일반 미분 관계가 제공됩니다.
주어진 컨벌루션에 대한 적분 속성은 임펄스 응답과 입력 응답의 컨벌루션을 생성합니다. 이것은 또한 반대 순서로 언급 될 수 있습니다.
스텝 함수로 컨볼루션을 수행하면 LTI 시스템은 이상적인 적분기처럼 동작합니다.

 합성곱 속성-ii

중요한 합성곱 속성에는 폭, 면적, 미분 및 적분 속성이 포함됩니다.

폭 속성은 입력 신호의 지속 시간이 T_1과 T_2인 경우 출력 응답의 폭은 두 함수의 모양과 관계없이 두 지속 시간의 합과 같음을 나타냅니다. 예를 들어, 지속 시간이 2초와 1초인 두 개의 직사각형 펄스를 합성하면 폭이 3초인 함수가 생성됩니다.

면적 속성은 두 함수의 합성곱 아래의 면적이 각 함수 아래의 면적의 곱과 같음을 나타냅니다. 수학적으로 x(t)와 ℎ(t)가 두 함수인 경우 면적 속성에 대한 표현식은 다음과 같이 주어질 수 있습니다.

<img alt="Equation1" src="/files/ftp_upload/16051/16051_PT%20EQUATION_1.svg" style="text-align: center; height: 16px;">

미분 속성은 합성곱의 미분이 입력 신호의 미분과 임펄스 응답의 합성곱과 같거나 그 반대임을 나타냅니다. 이는 다음과 같이 표현할 수 있습니다.

<img alt="Equation2" src="/files/ftp_upload/16051/16051_PT%20EQUATION_2.svg" style="text-align: center; height: 34px;">

이러한 방정식을 결합하고 고차 도함수에 대해 일반화하면 일반적인 미분 관계가 제공됩니다.

적분 속성은 합성곱의 적분이 임펄스 응답의 합성곱과 입력 신호의 적분을 생성하거나 그 반대임을 나타냅니다. 이는 수학적으로 다음과 같이 표현됩니다.

<img alt="Equation3" src="/files/ftp_upload/16051/16051_PT%20EQUATION_3.svg" style="text-align: center; height: 42px;">

합성곱이 계단 함수로 수행되는 경우 LTI 시스템은 이상적인 적분기처럼 작동합니다.

이러한 속성(폭, 면적, 미분, 적분)은 LTI 시스템에서 합성곱 연산을 단순화하고 이해하는 데 중요하며, 복잡한 신호 처리 작업을 분석하고 설계하는 것을 더 쉽게 만들어줍니다.

 합성곱 속성-II

중요한 컨볼루션 속성은 width, area, differentiation 및 integration 속성입니다.
width 속성은 지속 시간이 ...

Education

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JoVE Core

Electrical Engineering

전기공학

Unit 1

제1단원

Linear Time- Invariant Systems

선형 시불변 시스템

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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

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...

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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...

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

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...

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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 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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