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22The z-Transform

In Lecture 20, we developed the Laplace transform as a generalization of thecontinuous-time Fourier transform. In this lecture, we introduce the corre-

sponding generalization of the discrete-time Fourier transform. The resulting

transform is referred to as the z-transform and is motivated in exactly the

same way as was the Laplace transform. For example, the discrete-time Four-

ier transform developed out of choosing complex exponentials as basic build-

ing blocks for signals because they are eigenfunctions of discrete-time LTIsystems. A more general class of eigenfunctions consists of signals of the

form z, where z is a general complex number. A representation of discrete-

time signals with these more general exponentials leads to the z-transform.

As with the Laplace transform and the continuous-time Fourier trans-

form, a close relationship exists between the z-transform and the discrete-time Fourier transform. For z jn or, equivalently, for the magnitude of zequal to unity, the z-transform reduces to the Fourier transform. More gener-

ally, the z-transform can be viewed as the Fourier transform of an exponen-

tially weighted sequence. Because of this exponential weighting, the z-trans-

form may converge for a given sequence even if the Fourier transform does

not. Consequently, the z-transform offers the possibility of transform analysis

for a broader class of signals and systems.

As with the Laplace transform, the z-transform of a signal has associated

with it both an algebraic expression and a range of values of z referred to as

the region of convergence (ROC), for which this expression is valid. Two very

different sequences can have z-transforms with identical algebraic expres-

sions such that their z-transforms differ only in the ROC. Consequently, the

ROC is an important part of the specification of the z-transform.

Our principal interest in this and the following lectures is in signals fo rwhich the z-transform is a ratio of polynomials in z or in z 1.Transforms ofthis type are again conveniently described by the location of the poles roots

of the denominator polynomial) and the zeros roots of the numerator polyno-

mial) in the complex plane. The complex plane associated with the z-trans-

form is referred to as the z-plane. Of particular significance in the z-plane is

the circle of radius 1,concentric with the origin, referred to as the un t circle.

22-1



The z-Transform

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Signals and Systems

22-4

TRANSPARENCY22.1

The z-plane and the

unit circle.

TRANSPARENCY

22.2

The z-transform and

associated pole-zero

plot for a right-sided

exponential sequence.

Unit circle

Im

z p lane

Re

Z = ej



The z-Transform

22-5

TRANSPARENCY

22.3The z-transform and

associated pole-zero

plot for a left-sided

exponential sequence.

TRANSPARENCY

22.4Pole-zero plot for adiscrete-time under-

damped second-

order system

illustrating the

geometric determi-

nation of the Fourier

transform from the

pole-zero plot.



Signals and Systems

22-6

TRANSPARENCY

22.5Properties of the ROC

of the z-transform.

PROPERTIES OF THE REGION OF CONVERGENCE

. The ROC does not contain poles

. The ROC of X z) consists of a ring in the

z-plane centered about the origin

. y7x[n][converges <=>

. x [n] finite duration =>

ROC includes the unitcircle in the z-plane

ROC is entire z-planewith the possibleexception of z = 0 orz = oo

TRANSPARENCY

22.6Properties of the ROCfor a right-sided

sequence.

. Ix [n

N1 n

Sx [n] right-sided and IzI= r. is in ROC

=> all finite values of z for which izI> r0are in ROC

. x[n] right-sided and X z) rational=> RO isoutside the outermost pole

1. - -

0 0

ITITI



The z-Transform

22-

. x [n] left-sided and Iz = ro is in ROC=> all values of z for which 0 < IzI< r0will also be in ROC

. x[n] left-sided and X z) rational=> ROC inside the innermost pole

. x [n] two-sided and Iz = r is in ROC

=> ROC is a ring in the z-plane whichincludes the circle Iz|= rI

TRANSPARENCY22.7

Properties of the ROCof the z-transform for

a left-sided sequence

and for a two-sided

sequence.

TRANSPARENCY

22.8Transparencies 22.8-22.10 show ROCs for a

specified algebraic

expression for the z-

transform. Shown here

is the ROC if the

sequence is right-sided.



Signals and Systems

22-8

TRANSPARENCY

22.9The ROC if the

sequence is left-sided.

zx z) = _ _ _ _ _

(ZZ ) (Z- 2)

Im

Unit circle

z-plane

X-

TRANSPARENCY

22.10The ROC if the

sequence is two-sided.

zx z)= (z - Zz- 2

Im

Unit circle

z-plane

I--



The z-Transform

22-

MARKERBOARD22.2 (b)



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Resource: Signals and Systems Professor Alan V. Oppenheim

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