Name: convolution in lti systems and signal response
Uploaded: 2024-09-26T13:16:55.000Z
Duration: 1 min 24 s
Channel: JoVE

모든 LTI 시스템에서 두 신호의 컨볼루션은 모든 초기 조건이 0이라고 가정하고 컨볼루션 연산자를 사용하여 표시됩니다.
컨볼루션 적분은 제로 입력 또는 자연 응답과 시스템의 제로 상태 또는 강제 응답의 두 부분으로 나뉘며 t0은 초기 시간을 나타냅니다.
컨볼루션 적분은 입력 신호와 임펄스 응답이 모두 음의 시간 값에 대해 0이라고 가정하여 단순화할 수 있습니다.
그래픽 컨볼루션은 폴딩(Folding), 시프트(Shifting), 곱셈(Multiplication), 통합(Integration)의 4단계로 구성됩니다. 지정된 입력 펄스 신호와 출력 응답을 가진 RC 회로를 고려하십시오. 처음에 폴딩은 y축을 따라 입력 신호의 미러 이미지를 생성하여 수행됩니다. 그 다음에는 결과 신호의 약간의 이동, 곱셈 및 통합이 뒤따릅니다. 결과 회선은 그래픽으로 표시됩니다.
이산시간 컨볼루션에서 응답은 이산시간 시스템에 입력을 적용하고 임펄스 응답 및 컨볼루션 적분을 사용하여 결정됩니다.
이산 입력 신호와 임펄스 응답의 컨벌루션은 시스템 응답에 대한 컨볼루션 합을 형성합니다.

수학, 그래픽 및 이산 신호

모든 LTI(선형 시불변) 시스템에서 두 신호의 합성은 모든 초기 조건이 0이라고 가정하고 합성 연산자를 사용하여 표시합니다. 합성 적분은 두 부분으로 나눌 수 있습니다. 0 입력 또는 자연 응답과 0 상태 또는 강제 응답이며, t_0은 초기 시간을 나타냅니다.

합성 적분을 단순화하기 위해 입력 신호와 임펄스 응답이 모두 음의 시간 값에 대해 0이라고 가정합니다. 그래픽 합성 프로세스에는 폴딩, 이동, 곱셈 및 적분의 네 단계가 포함됩니다.

지정된 입력 펄스 신호와 출력 응답이 있는 RC 회로를 고려합니다. 처음에는 y축을 따라 입력 신호의 거울 이미지를 만들어서 폴딩을 수행합니다. 그 다음에 이동을 수행하여 폴딩 신호를 시간 축을 따라 슬라이드합니다. 다음으로, 폴딩 및 이동된 신호를 점별로 곱합니다. 마지막으로 결과 신호를 시간에 따라 적분하여 합성 결과를 제공합니다. 이 과정은 그래픽으로 표현할 수 있습니다.

이산 시간 합성에서 시스템의 응답은 이산 시간 시스템에 입력을 적용하고 임펄스 응답과 합성 합을 사용하여 결정됩니다. 이산 입력 신호 x[n]와 임펄스 응답 h[n]의 합성은 시스템 응답에 대한 합성 합을 형성합니다.

<img alt="Equation1" src="/files/ftp_upload/16049/16049_PT%20EQUATION_1.svg" style="text-align: center; height: 28px;">

이 합은 각 이산 시간 단계 n에서 출력 신호 y[n]를 계산합니다. 연속 합성과 이산 합성을 모두 이해하는 것은 LTI 시스템을 분석하고 다양한 입력에 대한 응답으로 동작을 예측하는 데 필수적입니다.

모든 LTI 시스템에서 두 신호의 컨볼루션은 모든 초기 조건이 0이라고 가정하고 컨볼루션 연산자를 사용하여 표시됩니다.
컨볼루션 ...

convolution-math-graphics-and-discrete-signals

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: 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 with, both the input signal and the impulse response are taken to be zero for negative time values.

Graphical convolution follows four steps: folding, shifting, multiplication, and integration. First, folding makes a mirror image of the input signal along the y-axis. Then the folded signal is shifted, or slid, along the time axis. After that, the folded and shifted signals are multiplied point by point.

The last step is integration over time, which gives the convolution result. This process can be shown graphically for an RC circuit with a given input pulse signal and output response. The same idea also appears in discrete-time systems. There, the input signal x[n] and the impulse response h[n] are combined with the convolution sum to find the output y[n] at each discrete time step n. Knowing both continuous and discrete convolution helps describe LTI system behavior for different inputs.

Convolution in LTI Systems and Signal Response

convolution-in-lti-systems-and-signal-response

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 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

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 convolution predicts LTI system response using folding, shifting, multiplication, and integration, plus the discrete convolution sum for x[n] and h[n].

Convolution in LTI Systems and Signal Response | 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 in LTI Systems and Signal Response

Description

Transcript

Tags