Name: z-transform rules for sum and convolution
Uploaded: 2024-09-26T13:17:10.000Z
Duration: 1 min 16 s
Channel: JoVE

La propriété d’accumulation est dérivée en exprimant la somme accumulée et en appliquant la propriété de décalage temporel pour résoudre la transformation Z.
Il stipule que la somme d’un signal à temps discret produit un autre signal dont la transformée Z est égale à la transformée Z du signal d’origine multipliée par z sur z moins 1.
La propriété de convolution montre que la convolution de deux signaux dans le domaine temporel aboutit au produit de leurs transformées Z dans le domaine fréquentiel.
Cela est valable pour les signaux causaux et non causaux.
L’application de la propriété de décalage temporel à l’équation du domaine temporel permet de vérifier la propriété de convolution.
Le théorème de la valeur initiale relie la valeur initiale d’un signal à sa transformée Z. Pour un signal x[n], la valeur initiale est la limite de X(z) lorsque z tend vers l’infini.
De même, le théorème de la valeur finale stipule que la valeur finale est la limite de 1 moins l’inverse de z multiplié par X(z) lorsque z tend vers un.
Elle ne s’applique que si x existe à l’infini et que tous les pôles sont à l’intérieur d’un cercle unitaire sauf en z égal à un.

propriétés de la transformée en z ii

La propriété d'accumulation dans le traitement du signal est dérivée en analysant la somme accumulée d'un signal à temps discret et en utilisant la propriété de décalage temporel pour déterminer sa transformée en z. Ce principe révèle que la transformée en z du signal additionné est liée à la transformée en z du signal d'origine par un facteur multiplicatif.

De plus, la propriété de convolution indique que la convolution de deux signaux dans le domaine temporel correspond au produit de leurs transformées en z dans le domaine fréquentiel. Cette propriété est valable pour les signaux causaux et non causaux. La propriété de convolution peut être confirmée en appliquant la propriété de décalage temporel à l'équation du domaine temporel correspondante.

Le théorème de la valeur initiale établit un lien entre la valeur initiale d'un signal et sa transformée en z. Pour un signal donné, la valeur initiale peut être obtenue en évaluant la transformée en z lorsque la variable s'approche de zéro. Ce théorème est particulièrement utile pour déterminer les conditions de départ d'un système à partir de sa transformée en z.

À l'inverse, le théorème de la valeur finale détermine la valeur finale d'un signal en examinant sa transformée en z lorsque la variable s'approche de 1. Ce théorème n'est applicable que si le signal continue d'exister à l'infini et que tous les pôles de la transformée en z se trouvent dans le cercle unité, sauf au point où la variable est égale à 1.

Ces propriétés sont cruciales pour l'analyse et la conception de systèmes à temps discret. En utilisant les théorèmes d'accumulation, de convolution, de valeur initiale et de valeur finale, le comportement des signaux à temps discret et des systèmes dans le domaine z peut être étudié efficacement. La maîtrise de ces propriétés permet la manipulation et la transformation des signaux, facilitant la création de filtres et de systèmes de contrôle qui fonctionnent dans le domaine du temps discret.

Propriétés de la transformée en Z II

La propriété d’accumulation est dérivée en exprimant la somme accumulée et en appliquant la propriété de décalage temporel pour résoudre la transformation Z.
Il indique ...

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Electrical Engineering

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z-Transform

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KEY TERMS AND CONCEPTS

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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 factor that links the two signals.

The convolution property shows that convolution in the time domain becomes multiplication of z-transforms in the frequency domain. This property applies to both causal signals, which depend on present and past values, and noncausal signals, which can also include future values. It can be verified by using the time-shifting property on the time-domain expression.

The initial value theorem connects a signal’s starting value to its z-transform. To find that initial value, evaluate the z-transform as the variable approaches zero. This is useful for finding the starting conditions of a system from its z-transform.

The final value theorem gives the ending value of a signal from its z-transform. It is found by examining the z-transform as the variable approaches one. This theorem works only if the signal still exists at infinity and the poles of the z-transform are inside the unit circle, except at the point where the variable equals one.

These properties are important for analyzing and designing discrete-time systems. Accumulation, convolution, the initial value theorem, and the final value theorem help describe signal behavior in the z-domain. They also support the design of filters and control systems that operate in the discrete-time domain.

Z-Transform Rules for Sum and Convolution

z-transform-rules-for-sum-and-convolution

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

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

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...

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 accumulation, convolution, and the initial and final value theorems connect discrete-time signals in the z-domain for system analysis and design.

Z-Transform Rules for Sum and 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

Z-Transform Rules for Sum and Convolution

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