Name: solving discrete-time responses with z-transform
Uploaded: 2024-09-26T13:17:11.000Z
Duration: 1 min 24 s
Channel: JoVE

A maioria dos sistemas práticos de tempo discreto pode ser representada por equações de diferença linear, tornando a transformada z uma ferramenta particularmente útil.
Conhecer o sinal de entrada e as N condições iniciais é necessário para resolver uma equação de diferença de N-ésima ordem.
Para sinais atrasados ou avançados, a transformada z desloca o sinal no domínio z, incluindo a multiplicação do inverso de z ou z, respectivamente.
Considere uma equação de diferença de segunda ordem caracterizada por coeficientes específicos e condições iniciais. A entrada é a função de passo unitário.
Tomando a transformada z de cada termo, a equação se transforma em uma expressão algébrica envolvendo a representação do domínio z dos sinais de entrada e saída.
Resolver esta equação algébrica para o sinal de saída do domínio z fornece uma expressão que pode ser simplificada usando a decomposição parcial da fração.
Os coeficientes são calculados e a resposta no domínio do tempo do sistema é dada pela transformada z inversa da expressão fracionária parcial.
Este processo demonstra o poder da transformada z na simplificação da análise e solução de sistemas lineares de tempo discreto, tornando-a uma ferramenta essencial em vários campos de processamento de sinais digitais e sistemas de controle.

solução de equação diferencial usando transformada z

A transformada z é uma ferramenta poderosa para analisar sistemas práticos de tempo discreto, frequentemente representados por equações de diferença lineares. Resolver uma equação de diferença de ordem superior requer conhecimento do sinal de entrada e das condições iniciais até um termo a menos que a ordem da equação.

A transformada z facilita o manuseio de sinais atrasados ​​ao deslocar o sinal no domínio z, o que corresponde a atrasar o sinal no domínio do tempo, e avançar sinais ao deslocar similarmente na direção oposta para um avanço de tempo.

Considere uma equação de diferença de segunda ordem com coeficientes específicos e condições iniciais, onde a entrada é uma função degrau unitário. Aplicar a transformada z a cada termo converte a equação de diferença em uma expressão algébrica no domínio z. Esta expressão envolve as representações do domínio z dos sinais de entrada e saída.

Para resolver o sinal de saída do domínio z, esta equação algébrica pode ser simplificada, frequentemente usando decomposição de fração parcial. Ao determinar os coeficientes para as frações parciais, obtemos uma forma gerenciável que pode ser invertida de volta ao domínio do tempo usando a transformada z inversa. A resposta resultante no domínio do tempo demonstra a eficácia da transformada z na simplificação da análise de sistemas lineares de tempo discreto. 


Este processo destaca a utilidade da transformada z em sistemas de controle e processamento de sinal digital. Ele fornece um método direto para a transição entre os domínios do tempo e z, resolver equações complexas e obter respostas precisas do sistema. É crucial considerar o papel das condições iniciais e da região de convergência ao aplicar a transformada z para garantir resultados precisos e significativos.

Solução de equação diferencial usando transformada Z

A maioria dos sistemas práticos de tempo discreto pode ser representada por equações de diferença linear, tornando a transformada z uma ferramenta particularmente útil.
Conhecendo o ...

difference-equation-solution-using-z-transform

Education

JoVE Core

Electrical Engineering

Engenharia Elétrica

Unit 1

Unidade 1

z-Transform

Transformada Z

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

Difference Equation Solution using z-Transform

solving discrete-time responses with z-transform

The z-transform helps solve discrete-time systems that are written as linear difference equations. These equations are common in digital signal processing and control systems. For a higher-order difference equation, you need the input signal and the initial conditions up to one term less than the order of the equation.

The method also makes delayed signals easier to handle. A delay in time becomes a shift in the z-domain. An advance in time is shifted in the opposite direction. This link between time shifts and z-domain shifts makes the algebra easier to work with.

A second-order difference equation with given coefficients and initial conditions can be converted step by step. In the example, the input is a unit step function. Applying the z-transform to each term turns the difference equation into an algebraic expression in the z-domain. That expression includes the z-domain forms of both the input and the output signals.

The algebraic equation for the output can then be simplified. Partial fraction decomposition is often used to break the result into smaller terms. After the coefficients are found, the expression can be converted back with the inverse z-transform. The final time-domain response shows how the z-transform simplifies analysis of discrete-time linear systems. The region of convergence must also be considered to make sure the result is accurate and meaningful.

Solving Discrete-Time Responses with z-Transform

solving-discrete-time-responses-with-z-transform

definition of z-transform

z-Transform for Discrete-Time Signals

The z-transform is a key math tool for analyzing discrete-time signals and systems. It is the discrete-time counterpart of the Laplace transform used for continuous-time systems. In signal and systems analysis, it works alongside the discrete-time Fourier transform. The z-transform changes a discrete-time signal into a power series with a complex variable, z. Each term in the series represents one sampling instant of the signal. This form makes it easier to study the signal in detail and solve...

region of convergence

Z-Transform Convergence and Stability

The z-transform and its Region of Convergence (ROC) are key ideas in discrete-time signals and systems. The z-transform is a mathematical tool for analyzing signals that change at separate time steps. Its values do not always converge, so the ROC identifies the range of complex numbers z where the transform is valid. The ROC can appear in different shapes in the complex plane. It may lie inside a circle, outside a circle, or within an annulus, which is the space between two circles. For an...

properties of the z-transform i

Z-Transform Rules for Signals

The z-transform rules for signals help students analyze discrete-time systems in digital signal processing. These rules turn time-domain changes into simpler expressions in the z-domain, which makes signal work easier to follow. Linearity is one key rule. It means the z-transform of a linear combination of two discrete-time signals equals the same linear combination of their individual z-transforms. This is useful when signals are added together or superimposed. Time-shifting is another...

properties of the z-transform ii

Z-Transform Rules for Sum and Convolution

The z-transform rules for sum and convolution help analyze discrete-time signals in signal processing. These rules show how changes in the time domain connect to the z-domain, which is the transform domain used for discrete-time systems. The accumulation property comes from the accumulated sum of a discrete-time signal. By applying the time-shifting property, the z-transform of the summed signal can be found from the z-transform of the original signal. The result is a simple multiplicative...

inverse z-transform by partial fraction expansion

inverse-z-transform-by-partial-fraction-expansion

Inverse z-Transform Using Poles and Residues

The inverse z-transform converts a function from the z-domain back to the time domain. One useful approach is the partial fraction method. It breaks a complex function into simpler fractions with distinct coefficients. The process starts by identifying the poles of the function. A pole is a value that makes the function grow without a finite limit. The function is then written in terms of these poles, and each pole adds one term to the partial fraction form. Next, the coefficients for each...

difference equation solution using z-transform

The z-transform helps solve discrete-time systems that are written as linear difference equations. These equations are common in digital signal processing and control systems. For a higher-order difference equation, you need the input signal and the initial conditions up to one term less than the order of the equation. The method also makes delayed signals easier to handle. A delay in time becomes a shift in the z-domain. An advance in time is shifted in the opposite direction. This link...

relation of dft to z-transform

DFT Through the Unit Circle

The Discrete Fourier Transform (DFT) shows how a discrete-time signal breaks into frequency parts. It takes N samples from the time domain and gives N values in the frequency domain. Each value represents one frequency component. The z-transform helps explain this process. A z-transform represents a discrete sequence in the complex frequency domain by summing sequence terms multiplied by powers of a complex number. For causal sequences, which start at a fixed time and move forward, the sum can...

key terms and concepts

Learn how the z-transform turns a difference equation into algebra, handles time shifts, and helps find the discrete-time response with initial conditions.

Solving Discrete-Time Responses with z-Transform | 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

Solving Discrete-Time Responses with z-Transform

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