### 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 language | English (US) |
---|---|

Title of host publication | Fundamentals of Circuits and Filters |

Publisher | CRC Press |

Pages | 5-1-5-17 |

ISBN (Electronic) | 9781420058888 |

ISBN (Print) | 1420058878, 9781420058871 |

State | Published - Jan 1 2009 |

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### ASJC Scopus subject areas

- Engineering(all)
- Computer Science(all)

### Cite this

*Fundamentals of Circuits and Filters*(pp. 5-1-5-17). CRC Press.

**z-transform.** / Kovacevic, Jelena.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

}

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 -