Name: inverse z-transform using poles and residues
Uploaded: 2024-09-26T13:17:10.000Z
Duration: 1 min 20 s
Channel: JoVE

La trasformata Z inversa è uno strumento essenziale utilizzato per convertire una funzione dalla sua rappresentazione nel dominio della frequenza al dominio del tempo.
Si consideri la funzione X(z), che deve essere riconvertita nella sua rappresentazione nel dominio del tempo.
Per scomporre X(z), si identificano i poli della funzione, che viene espressa in termini di questi poli.
Ogni polo contribuisce con un termine all'espansione della frazione parziale.
I coefficienti per ogni termine nell'espansione sono determinati sostituendo valori specifici per z.
Dopo aver determinato tutti i coefficienti, la funzione viene riassemblata nella sua forma scomposta.
Questa nuova rappresentazione è più gestibile.
Ogni frazione corrisponde a coppie di trasformate Z note, rendendo semplice la trasformazione inversa.
Il metodo delle frazioni parziali è una tecnica efficace per trovare la trasformata Z inversa scomponendo una funzione in frazioni più semplici con coefficienti distinti.
La trasformata Z inversa viene applicata separatamente a ciascun termine frazionario, risultando in una combinazione di funzioni delta, sequenze esponenziali e funzioni a gradini, che rappresentano collettivamente la sequenza originale nel dominio del tempo.

trasformata z inversa tramite espansione di frazioni parziali

Inverse z-Transform by Partial Fraction Expansion

La trasformata z inversa è una tecnica fondamentale per convertire una funzione dalla sua rappresentazione del dominio z al dominio del tempo. Un metodo efficace per trovare la trasformata z inversa è quello delle frazioni parziali, che prevede la scomposizione di una funzione in frazioni più semplici con coefficienti distinti. Queste frazioni corrispondono a coppie di trasformate z note, facilitando il processo di trasformazione inversa.


Per iniziare il processo, vengono identificati i poli della funzione e la funzione viene espressa in termini di questi poli. Ogni polo contribuisce con un termine alla scomposizione della frazione parziale. I coefficienti per ogni termine nella scomposizione vengono determinati valutando i residui in ogni polo.


Una volta determinati i coefficienti, la funzione viene riassemblata nella sua forma scomposta, rendendola più semplice da usare. La trasformata z inversa viene quindi applicata a ogni termine frazionario separatamente. Il risultato combina le funzioni delta, le sequenze esponenziali e le funzioni a gradini che rappresentano la sequenza originale del dominio del tempo.


Attraverso il Partial Fraction Method, la trasformata z inversa delle funzioni complesse diventa più gestibile, consentendo una conversione accurata nel dominio del tempo. Questo metodo si assicura che ogni componente della funzione decomposta venga trasformata correttamente, con una conseguente ricostruzione precisa della sequenza originale.

Trasformata Z inversa tramite espansione di frazioni parziali

La trasformata Z inversa è uno strumento essenziale utilizzato per convertire una funzione dalla sua rappresentazione nel dominio della frequenza al dominio del tempo.
Considera il ...

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

Education

Didattica

JoVE Core

Electrical Engineering

Ingegneria elettrica

Unit 1

Unità 1

z-Transform

Trasformata Z

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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 term are found by evaluating the residues at each pole. The decomposed terms are then matched with known z-transform pairs. The inverse z-transform is applied to each fractional term separately.

The final result combines delta functions, exponential sequences, and step functions. Together, these terms reconstruct the original time-domain sequence. The partial fraction method makes complex inverse z-transforms easier to handle and supports accurate conversion back to the time domain.

Inverse z-Transform Using Poles and Residues

inverse-z-transform-using-poles-and-residues

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

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 the inverse z-transform uses poles, residues, and partial fraction decomposition to turn a z-domain function back into a time-domain sequence.

Inverse z-Transform Using Poles and Residues | 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

Inverse z-Transform Using Poles and Residues

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