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

ASJC Scopus subject areas

  • Engineering(all)
  • Computer Science(all)

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  • Cite this

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