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

Em qualquer sistema LTI, a convolução de dois sinais é denotada usando um operador de convolução, assumindo que todas as condições iniciais são zero.
A integral de convolução é dividida em duas partes: a entrada zero ou resposta natural e o estado zero ou resposta forçada do sistema, com t0 indicando o tempo inicial.
A integral de convolução pode ser simplificada presumindo que tanto o sinal de entrada quanto a resposta ao impulso são zero para valores de tempo negativos.
A convolução gráfica envolve quatro etapas: dobra, deslocamento, multiplicação e integração. Considere um circuito RC com um sinal de pulso de entrada especificado e resposta de saída. Inicialmente, o dobramento é realizado criando uma imagem espelhada do sinal de entrada ao longo do eixo y. Isso é seguido por uma ligeira mudança, multiplicação e integração dos sinais resultantes. A convolução resultante é representada graficamente.
Na convolução em tempo discreto, a resposta é determinada pela aplicação de uma entrada a um sistema de tempo discreto e usando a resposta ao impulso e a integral de convolução.
A convolução do sinal de entrada discreto e a resposta ao impulso formam a soma da convolução para a resposta do sistema.

matemática, gráficos e sinais discretos

Em qualquer sistema LTI (Linear Time-Invariant, ou linear invariante no tempo ), a convolução de dois sinais é denotada usando um operador de convolução, assumindo que todas as condições iniciais são zero. A integral de convolução pode ser dividida em duas partes: a entrada zero ou resposta natural e o estado zero ou resposta forçada, com t_0 indicando o tempo inicial.
Para simplificar a integral de convolução, assume-se que tanto o sinal de entrada quanto a resposta ao impulso são zero para valores de tempo negativos. O processo de convolução gráfica envolve quatro etapas: dobramento, deslocamento, multiplicação e integração.
Considere um circuito RC com um sinal de pulso de entrada especificado e resposta de saída. Inicialmente, o dobramento é realizado criando uma imagem espelhada do sinal de entrada ao longo do eixo y. Isso é seguido pelo deslocamento, onde o sinal dobrado é deslizado ao longo do eixo do tempo. Em seguida, a multiplicação dos sinais dobrados e deslocados é feita ponto a ponto. Finalmente, a integração do sinal resultante ao longo do tempo fornece o resultado da convolução. Este processo pode ser representado graficamente.
Na convolução de tempo discreto, a resposta do sistema é determinada aplicando uma entrada a um sistema de tempo discreto e usando a resposta ao impulso e a soma da convolução. A convolução do sinal de entrada discreto x[n] e a resposta ao impulso h[n] formam a soma da convolução para a resposta do sistema:
<img alt="Equation1" src="/files/ftp_upload/16049/16049_PT%20EQUATION_1.svg" style="height: 28px;">
Esta soma calcula o sinal de saída y[n] em cada passo de tempo discreto n. Entender a convolução contínua e discreta é essencial para analisar sistemas LTI e prever seu comportamento em resposta a várias entradas.

Matemática, Gráficos e Sinais Discretos

Em qualquer sistema LTI, a convolução de dois sinais é denotada usando um operador de convolução, assumindo que todas as condições iniciais são zero.
A convolução ...

convolution-math-graphics-and-discrete-signals

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JoVE Core

Electrical Engineering

Engenharia Elétrica

Unit 1

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Linear Time- Invariant Systems

Sistemas Lineares Invariantes no Tempo

KEY TERMS AND CONCEPTS

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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

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