z-transform

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

When analyzing linear systems, one of the problems we often encounter is that of solving linear, constant-coefficient differential equations. A tool used for solving such equations is the Laplace transform. At the same time, to aid the analysis of linear systems, we extensively use Fourier-domain methods. With the advent of digital computers, it has become increasingly necessary to deal with discrete-time signals, or, sequences. These signals can be either obtained by sampling a continuous-time signal, or they could be inherently discrete. To analyze linear discrete-time systems, one needs a discrete-time counterpart of the Laplace transform (LT). Such a counterpart is found in the z-transform, which similarly to the LT, can be used to solve linear constant-coefficient difference equations. In other words, instead of solving these equations directly, we transform them into a set of algebraic equations first, and then solve in this transformed domain. On the other hand, the z-transform can be seen as a generalization of the discrete-time Fourier transform (FT) X(ejv) 1/4 XÞ1 n1/41 x[n]ejvn (5:1) This expression does not always converge, and thus, it is useful to have a representation which will exist for these nonconvergent instances. Furthermore, the use of the z-transform offers considerable notational simplifications. It also allows us to use the extensive body of work on complex variables to aid in analyzing discrete-time systems.

Original languageEnglish (US)
Title of host publicationFundamentals of Circuits and Filters
PublisherCRC Press
Pages5-1-5-17
ISBN (Electronic)9781420058888
ISBN (Print)1420058878, 9781420058871
StatePublished - Jan 1 2009

Fingerprint

Laplace transforms
Mathematical transformations
Linear systems
Difference equations
Digital computers
Fourier transforms
Differential equations
Sampling

ASJC Scopus subject areas

  • Engineering(all)
  • Computer Science(all)

Cite this

Kovacevic, J. (2009). z-transform. In Fundamentals of Circuits and Filters (pp. 5-1-5-17). CRC Press.

z-transform. / Kovacevic, Jelena.

Fundamentals of Circuits and Filters. CRC Press, 2009. p. 5-1-5-17.

Research output: Chapter in Book/Report/Conference proceedingChapter

Kovacevic, J 2009, z-transform. in Fundamentals of Circuits and Filters. CRC Press, pp. 5-1-5-17.
Kovacevic J. z-transform. In Fundamentals of Circuits and Filters. CRC Press. 2009. p. 5-1-5-17
Kovacevic, Jelena. / z-transform. Fundamentals of Circuits and Filters. CRC Press, 2009. pp. 5-1-5-17
@inbook{654d5c134ef3407788c49ed680ec0d56,
title = "z-transform",
abstract = "When analyzing linear systems, one of the problems we often encounter is that of solving linear, constant-coefficient differential equations. A tool used for solving such equations is the Laplace transform. At the same time, to aid the analysis of linear systems, we extensively use Fourier-domain methods. With the advent of digital computers, it has become increasingly necessary to deal with discrete-time signals, or, sequences. These signals can be either obtained by sampling a continuous-time signal, or they could be inherently discrete. To analyze linear discrete-time systems, one needs a discrete-time counterpart of the Laplace transform (LT). Such a counterpart is found in the z-transform, which similarly to the LT, can be used to solve linear constant-coefficient difference equations. In other words, instead of solving these equations directly, we transform them into a set of algebraic equations first, and then solve in this transformed domain. On the other hand, the z-transform can be seen as a generalization of the discrete-time Fourier transform (FT) X(ejv) 1/4 X{\TH}1 n1/41 x[n]ejvn (5:1) This expression does not always converge, and thus, it is useful to have a representation which will exist for these nonconvergent instances. Furthermore, the use of the z-transform offers considerable notational simplifications. It also allows us to use the extensive body of work on complex variables to aid in analyzing discrete-time systems.",
author = "Jelena Kovacevic",
year = "2009",
month = "1",
day = "1",
language = "English (US)",
isbn = "1420058878",
pages = "5--1--5--17",
booktitle = "Fundamentals of Circuits and Filters",
publisher = "CRC Press",

}

TY - CHAP

T1 - z-transform

AU - Kovacevic, Jelena

PY - 2009/1/1

Y1 - 2009/1/1

N2 - When analyzing linear systems, one of the problems we often encounter is that of solving linear, constant-coefficient differential equations. A tool used for solving such equations is the Laplace transform. At the same time, to aid the analysis of linear systems, we extensively use Fourier-domain methods. With the advent of digital computers, it has become increasingly necessary to deal with discrete-time signals, or, sequences. These signals can be either obtained by sampling a continuous-time signal, or they could be inherently discrete. To analyze linear discrete-time systems, one needs a discrete-time counterpart of the Laplace transform (LT). Such a counterpart is found in the z-transform, which similarly to the LT, can be used to solve linear constant-coefficient difference equations. In other words, instead of solving these equations directly, we transform them into a set of algebraic equations first, and then solve in this transformed domain. On the other hand, the z-transform can be seen as a generalization of the discrete-time Fourier transform (FT) X(ejv) 1/4 XÞ1 n1/41 x[n]ejvn (5:1) This expression does not always converge, and thus, it is useful to have a representation which will exist for these nonconvergent instances. Furthermore, the use of the z-transform offers considerable notational simplifications. It also allows us to use the extensive body of work on complex variables to aid in analyzing discrete-time systems.

AB - When analyzing linear systems, one of the problems we often encounter is that of solving linear, constant-coefficient differential equations. A tool used for solving such equations is the Laplace transform. At the same time, to aid the analysis of linear systems, we extensively use Fourier-domain methods. With the advent of digital computers, it has become increasingly necessary to deal with discrete-time signals, or, sequences. These signals can be either obtained by sampling a continuous-time signal, or they could be inherently discrete. To analyze linear discrete-time systems, one needs a discrete-time counterpart of the Laplace transform (LT). Such a counterpart is found in the z-transform, which similarly to the LT, can be used to solve linear constant-coefficient difference equations. In other words, instead of solving these equations directly, we transform them into a set of algebraic equations first, and then solve in this transformed domain. On the other hand, the z-transform can be seen as a generalization of the discrete-time Fourier transform (FT) X(ejv) 1/4 XÞ1 n1/41 x[n]ejvn (5:1) This expression does not always converge, and thus, it is useful to have a representation which will exist for these nonconvergent instances. Furthermore, the use of the z-transform offers considerable notational simplifications. It also allows us to use the extensive body of work on complex variables to aid in analyzing discrete-time systems.

UR - http://www.scopus.com/inward/record.url?scp=85057449896&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85057449896&partnerID=8YFLogxK

M3 - Chapter

SN - 1420058878

SN - 9781420058871

SP - 5-1-5-17

BT - Fundamentals of Circuits and Filters

PB - CRC Press

ER -