Getting Started with Engineering
Cover of Getting Started with Engineering book with cartoon boy and girl in yellow shirts.
Explore Book
Buy NowSubscribe on Perlego
Getting Started with Engineering
Cover of Getting Started with Engineering book with cartoon boy and girl in yellow shirts.Explore Book
Buy NowSubscribe on Perlego

The z-transform (ZT) is a generalization of the discrete-time Fourier transform (DTFT) for discrete-time signals, but the ZT applies to a broader class of signals than the DTFT. The two-sided or bilateral z-transform (ZT) of sequence x[n] is defined as

image0.jpg

The ZT operator transforms the sequence x[n] to X(z), a function of the continuous complex variable z. The relationship between a sequence and its transform is denoted as

image1.jpg

You can establish the connection between the discrete-time Fourier transform (DTFT) and the ZT by first writing

image2.jpg

The special case of r = 1 evaluates X(z) over the unit circle —

image3.jpg

and is represented as

image4.jpg

the DTFT of x[n]. This result holds as long as the DTFT is absolutely summable (read: impulse functions not allowed).

The view that

image5.jpg

sampled around the unit circle in the z-plane

image6.jpg

shows that the DTFT has period 2π because

image7.jpgimage8.jpgimage9.jpg

About This Article

This article is from the book: 

About the book author:

Mark Wickert, PhD, is a Professor of Electrical and Computer Engineering at the University of Colorado, Colorado Springs. He is a member of the IEEE and is doing real signals and systems problem solving as a consultant with local industry.