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

컨볼루션 계산은 고유한 속성을 활용하여 단순화할 수 있습니다.
commutative 속성은 LTI 시스템의 입력과 임펄스 응답이 출력에 영향을 주지 않고 상호 교환될 수 있음을 나타냅니다.
associative 속성은 세 함수의 병합된 컨볼루션이 컨볼루션이 실행되는 순서에 관계없이 변경되지 않은 상태로 유지됨을 시사합니다.
임펄스 응답을 가진 두 개의 LTI 시스템이 직렬로 연결되면 연관 속성을 사용하여 각각의 방정식을 결합하여 개별 임펄스 응답의 컨볼루션과 유사한 등가 조인트 임펄스를 도출할 수 있습니다.
distributive 속성은 여러 입력 신호의 합에 대한 컨볼루션 연산을 가능하게 하고 복잡한 임펄스 응답을 더 간단한 구성 요소로 분해하여 단순화합니다.
time-shift 속성은 시불변 시스템의 입력을 지연시키거나, 시스템에 지연이 내장되어 있는 경우 두 지연의 합만큼 출력이 지연됨을 의미합니다.
계산적으로 이 속성을 사용하면 대칭 또는 인과 관계를 활용하기 위해 신호를 지연시키거나 진행할 수 있으므로 컨볼루션 연산을 단순화할 수 있습니다.

합성곱 특성-i

합성곱 연산은 고유 특성을 활용하여 간소화할 수 있습니다.

교환 법칙은 LTI(선형 시불변) 시스템의 입력과 임펄스 응답을 출력에 영향을 주지 않고 바꿀 수 있음을 보여줍니다.

<img alt="Equation1" src="/files/ftp_upload/16050/16050_PT%20EQUATION_1.svg" style="text-align: center; height: 16px;">

결합 법칙은 세 함수의 병합 합성곱이 합성곱 순서에 관계없이 변경되지 않음을 시사합니다. 예를 들어, 세 함수 x(t), ℎ_1(t), ℎ_2(t)는 다음과 같이 작성됩니다.

<img alt="Equation2" src="/files/ftp_upload/16050/16050_PT%20EQUATION_2.svg" style="text-align: center; height: 16px;">

임펄스 응답이 있는 두 개의 LTI 시스템이 직렬로 연결된 경우, 각각의 방정식을 결합 법칙을 사용하여 결합하여 동등한 결합 임펄스 응답을 도출할 수 있습니다. 이는 개별 임펄스 응답의 합성곱입니다.

분배 속성은 여러 입력 신호의 합에 대한 합성 연산을 가능하게 하여 복잡한 임펄스 응답을 더 간단한 구성 요소로 분해할 수 있습니다. 수학적으로 이는 다음과 같이 표현됩니다.

<img alt="Equation3" src="/files/ftp_upload/16050/16050_PT%20EQUATION_3.svg" style="text-align: center; height: 16px;">

시간 이동 속성은 시간 불변 시스템의 입력을 지연하면 출력이 같은 양만큼 지연됨을 의미합니다. 마찬가지로 시스템에 내장된 지연이 있는 경우 출력은 입력 지연과 시스템 지연의 합만큼 지연됩니다. 시간 이동 t_0의 경우:

<img alt="Equation4" src="/files/ftp_upload/16050/16050_PT%20EQUATION_4.svg" style="text-align: center; height: 16px;">

계산적으로 이 속성은 신호를 지연하거나 진행할 수 있게 하여 대칭성 또는 인과성을 활용하여 합성 연산을 간소화합니다.

이러한 속성(가환, 결합, 분배 및 시간 이동)은 LTI 시스템에서 합성 연산을 간소화하여 복잡한 신호 처리 작업을 보다 관리하기 쉽고 효율적으로 만드는 기본 도구입니다.

합성곱 특성-I

컨볼루션 계산은 고유한 속성을 활용하여 단순화할 수 있습니다.
교환적 속성은 입력과 임펄스가 ...

Education

교육

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 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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