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22.10 Properties of the z transform 709
jv
s plane
8p
3 a 0 + —– j T
6p
3 a 0 + —– j T
4p
3 a 0 + —– j T
2p
3 a 0 + —– j T
3 a 0
s
jv
s plane jy z plane
2p
3 a 0 – —– j T
4p
3 a 0 – —– j T
6p
3 a 0 – —– j T
8p
3 a 0 – —– j T
Figure22.19
The effect ofsampling isto introduce an
infinite set ofpoles.
s
Figure22.20
Theleft half ofthesplane maps to the inside ofthe unit
circle ofthezplane.
1
1
x
The great advantage of thezplane is that it eliminates the problem of infinitely repeatingpolesandzeroswhenanalysingdiscretesystems.Thiscanbeillustratedbyconsideringhow
the imaginary axis of thesplane maps tothezplane.
Referring to Figure 22.20, we see that s = σ + jω. On the imaginary axis σ = 0
andthereforez = e sT = e jωT .As ω variesbetween − π T and π T ,thelocusofzisacircle
of radius 1, centred at the origin (see Section 9.10). As ω is increased from 0 to π T the
upper half of the unit circle is traced out, while as ω is decreased from 0 to − π T , the
lower half of the unit circle is traced out. Increasing ω above π T
or decreasing it below
− π T leadstoaretracingoftheunitcircle.Inotherwords,therepeatedsplanepointsare
superimposedontopofeachother.Thisisthereasonwhythezplaneapproachismuch
simplerthan thesplane approach when analysing discrete systems.
Theztransforms of discrete signals and systems are, in many cases, simple ratios of
polynomials. We shall see shortly that this means the process of analysing difference
equations which model these signals and systems is reduced to relatively simple algebraicmanipulations.
22.10 PROPERTIESOFTHEzTRANSFORM
Because of the relationship between the two transforms we would expect that many of
the properties of the Laplace transform would be mirrored by properties of theztransform.
This isindeedthe caseand someoftheseproperties aregiven now. These are:
(1) linearity;
(2) shift theorems;
(3) the complex translation theorem.