Wiley signals and systems e book TLFe BO 334

319

13.1. Definition and Examples

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Figure 13.5: Region of convergencc for the fiiriction X(z) from Kxnmplp 13.Y

region of tmivrrgencc is cirrular licrc too, and its iadius is equal to tlie mngiiitiide
of the pole.
In ccrmptirison to Examplc 13.2, wc irotice that the z-transform of thr riglit sided cxponential sequence (13A) arid the left-sided cxponmtiaI sequence (13.7)
h v e the sarnt’ form and only thc regions of convergerice axe diffeierrt (Figiws 13.3
the region of corivergencc.
and 13.5). This eiriph
M’ithout it. a unique
IVc are familiar wiih this sit uation from the l ~ p l a c ctransform. In Exainp l ~ s4.1 arid 4.2 we corisiclcred lcft-sided mid right-sidtvi continuous-time signals
tlint arc’ likewise orily ( ~ s t i ~ ~ ~ iby
~ ithe
s ~ region
i t ~ t ~of corivergence (see Fignres 4.3
arid 4.4).

ne
Wi. cari iriterprct the individual points of the :-plane in a similar way t o the splane in Chapter 3.1.3. ti’ignrc 13.6 shon s the corresponding exponential secpen(*+’
zk for different values of z .
Tht. valucs z = eJb’ on t,lie w i t circle correspond to thc cqtonential serim
eJc’kwitlicorrstant amplitude: z = 1 lei& to a series with constant values because
eJok = l k =; 1, whilc z = - 1 is the highest repiesentable frequency, 1wc.ai.w
F ) x k = (- J ) k . All other valiieq cm thc unit circle repr
rit complex expotieni in1
oscillations ctf frcqiicncy 6 1 with --?r < i

l ic T . c‘omplcx coiijngate values oi 3
a r t distingms2ied by the tlirectioii of rotation. Values of z = I’c~‘‘witlirri ihc unit
circle (/. < 1) b ~ l o n gto a decaying exltoncntial stqucnce arid vdiies out.;idc the
unit circlc ( T 2 1) belong t o c? growiiig exponelitis1 seqiieiice. In Figure 13.6 the
exponential srries z h are each oiily slrom7n for k 2 0. This shooltf not lead t o the
misintcrprelatiori that we are onlv dcalirig wit11 rnii1;uteral scquences. since d s o for
k < 0,z k f 0.